ARITHMETIC 


RITTER. 


QA 

135 

R51 


Hitter., 


L 


Southern  Branch 
of  the 

University  of  California 

Los  Angeles 


Form  L-l 


This  book  is  DUE  on  the  last  date  stamped  below 


281 


1925 


JUL 


form  L-9-15»i-8,'24 


PEDAGOGICS 


APPLIED  TO- 


ARITHMETIC. 


— BY — 


PROFESSOR  OF  MATHEMATICS  IN  THE  STATE  NORMAL  SCHOOL,  CHICO,  GAL. 


"HE  THEREFORE  THAT  IS  ABOUT  CHILDREN  SHOULD  WELL 
STUDY  THEIR  NATURES  AND  APTITUDES,  AND  SEE  BY  OFTEN 
TRIALS  WHAT  TURN  THEY  EASILY  TAKE,  AND  WHAT  BECOMES 
THEM  ;  OBSERVE  WHAT  THEIR  STOCK  IS,  HOW  IT  MAY  BE 
IMPROVED,  AND  WHAT  IT  IS  FIT  FOR." — Locke. 


STOCKTON,  CAL. 

LEROY  S.   ATWOOD,   PRINTER. 

1891. 


61233 


COPYRIGHT,  1891, 
BY  C.   M.   HITTER. 


QA 
13S" 


CONTENTS. 


Page 

PREFACE 7 

PART    I.— THE  SUPPLY 11 

THE  BEGINNING  OF  NUMBER  WORK 13 

1  TO  10 13 

TENS,  10  TO  100 26 

10  TO  20 36 

20  TO  100 43 

100  TO  1000 49 

1000  AND   ABOVE 53 

THE  PRIME  NUMBER 58 

G.  C.   D.  AND  L.  C.   M 61 

FRACTIONS 66 

FRACTIONS — DECIMALS 75 

DECIMALS — PER  CENT 78 

TABULATIONS 81 

PART  II.— THE  DEMAND 103 

PERCENTAGE — PROFIT  AND  Loss 105 

BUSINESS  DISCOUNT 107 

INSURANCE 108 

COMMISSION 100 

TAXES Ill 

STOCKS 113 

INTEREST 115 

BANKING 120 

LONGITUDE  AND  TIME 124 

THE  METRIC  SYSTEM 133 

THE  PUBLIC  LANDS 137 

AVERAGE  OF  ACCOUNTS 144 

PROPORTION 147 

GENERAL  AVERAGE — SHIPPING 149 

ALLIGATION 150 

MENTAL  DISCIPLINE 151 

RESUME...  ..159 


PREFACE. 


The  object  that  I  have  had  in  view  in  the  preparation  of  this 
work  has  been  the  better  teaching  of  that  branch  of  mathematics 
whose  utility  no  one  questions.  Among  the  masses  of  the 
American  people  arithmetic  has  been  regarded  of  more  import- 
ance than  any  other  study  in  the  educational  curriculum.  Other 
studies  must  be  neglected,  if  necessary,  in  order  that  it  may  be 
mastered.  This  state  of  affairs  being  admitted,  as  well  as  the  great 
evil  that  would  result  from  the  neglect  of  other  most  desirable 
branches  of  study,  it  seems  to  be  the  duty  of  those  most  con- 
versant with  the  temper  of  our  people  and  the  time  devoted  to 
arithmetic,  to  put  forth  every  energy  to  lessen  the  time,  increase 
the  efficiency,  and  minimize  the  useless  efforts  that  attach  to 
the  teaching  and  acquiring  of  this  subject.  It  is  certain  that 
no  more  fertile  field  is  open  in  which  teachers  may  reap  ap- 
preciation for  their  untiring  energies;  for,  as  Agassiz  has  said, 
' '  On  the  broad  high  road  of  civilization  along  which  men  are 
ever  marching,  they  pass  by  unnoticed  the  land  marks  of  intel- 
lectual progress,  unless  they  chance  to  have  some  direct  bearing 
on  what  is  called  the  practical  side  of  life. ' ' 

An  experience  of  eighteen  years  in  teaching  in  country  schools, 
in  city  grammar  and  high  schools,  and  in  the  State  Normal 
School,  has  enabled  me  to  see  and  to  feel  the  benefits  of  good 
methods  and  the  evils  of  poor  ones.  Method  is  not  arbitrary 
but  rational,  not  inflexible  but  natural;  and  he  that  profits  the 
most  by  suggestions  and  by  aids  is  not  he  that  literally  follows, 
but  he  that  digests  and  assimilates.  Rosencranz  says:  "The 
peculiarities  of  the  person  who  is  to  be  educated  and,  in  fact, 
all  the  existing  circumstances  necessitate  an  adaptation  of  the 
universal  aims  and  ends,  that  cannot  be  provided  for  before 


8  PREFACE. 

hand,  but  must  rather  test  the  ready  tact  of  the  educator  who 
knows  how  to  take  advantage  of  the  existing  conditions  to 
fulfill  his  desired  end."  Hence  I  have  aimed,  not  specifically  to 
put  in  the  mouth  of  the  teacher  words  and  formulae  which  he 
shall  brainlessly  and  heartlessly  utter  in  the  pupils'  hearing,  but 
rather  general  ideas  embodied  in  formulations  that  shall  be 
suggestive  to  the  receptive  mind,  that  shall  enable  the  true 
teacher  to  see  his  own  faults  and  correct  them,  and  that  shall 
open  up  new  fields  for  growth. 

The  underlying  principle  of  education  is  the  self  activity  of  the 
pupil's  mind.  For  the  impressment  of  entirely  new  ideas  the 
monologue  is  conceived  to  be  the  natural  method  of  teaching; 
for  the  enlargement  of  the  thought,  the  fixing  of  a  concept,  and 
the  preparation  of  the  mind  for  a  new  principle,  the  dialogue  is 
believed  to  be  far  preferable.  The  aids  are  objects  in  the  hands 
of  both  the  teacher  and  the  pupil;  many  and  varied  in  the  hands 
of  the  small  child  and  gradually  decreasing  in  number  and 
variety  as  the  mind  of  the  child  becomes  the  storehouse  of 
faultless  concepts  drawn  from  objective  percepts.  While  coun- 
seling against  eccentricity,  I  would,  in  the  interest  of  a  live 
school  and  a  progressive  teacher  commend  the  following  from 
Tate:  "As  children  love  change  and  novelty,  a  good  teacher 
will  vary  his  subjects  of  instruction  as  well  as  his  methods  of 
instruction  accordingly;  his  judgment  must  be  exercised  in 
selecting  those  methods  which  are  most  suited  to  the  existing 
conditions  of  his  school." 

As  the  work  in  numbers  progresses  and  when  all  that  is 
fundamental,  the  simple  number  and  the  fraction  (with  its  varia- 
tions, decimals  and  per  cent.),  have  been  thoroughly  compre- 
hended by  the  pupils,  the  hand  of  the  teacher  should  rest  more 
and  more  lightly  upon  the  pupil;  hence  the  pupils  are  encouraged 
to  make  original  investigations,  under  direction  of  the  teacher 
at  first,  into  such  subjects  as  Commission,  Banking,  and  Taxes, 
and  report  for  class  consideration  what  they  have  learned. 
Harris  says:  "All  teachers  must  keep  in  view  the  standpoint 
of  the  pupil,  use  illustration,  and  supply  necessary  steps  to  make 
the  connection  clear  to  the  pupil.  The  live  teacher  is  careful 
to  avoid  being  hampered  by  the  limits  of  any  one  method, 
although  he  finds  use  for  all  on  occasions. ' ' 

It  is  suggested  that  the  teacher  in  making  use  of  the  methods 


PREFACE. 

herein  outlined  keep  in  mind  that  a  second  step  should  not  be 
taken  until  the  first  has  been  thoroughly  made;  to  that  end  it 
will  be  found  necessary  to  multiply  explanations,  questions, 
and  exercises,  and  to  vary  the  same  as  much  as  possible. 

In  written  or  blackboard  work,  the  end  sought  should  be 
short  solutions  and  clear  oral  explanations.  This  end  is  most 
readily  reached  by  having  the  pupils  compare  and  criticise  both 
the  solutions  and  the  explanations.  This  exercise  awakens 
renewed  interest,  and  is  always  a  feature  of  a  well  conducted 
school. 

The  arrangement  or  order  of  presentation  of  subjects,  it  is 
thought,  will  commend  itself  as  being  based  upon  psychological 
principles.  The  supply  precedes  the  demand. 

It  is  assumed  that  the  teacher  aims  to  be  as  useful  as  possible. 
To  be  as  useful  as  possible  he  must  be  progressive.  He  must 
commingle  with  the  world  with  his  eyes  and  ears  open.  He 
must  be  sociable.  He  must  be  high-minded,  honest,  truthful, 
and  moral  in  all  respects.  He  must  not  forget  that  he  is  the 
cynosure  of  the  school  and  of  the  community.  He  must  read 
educational  periodicals,  and  must  study  psychological  and  peda- 
gogical literature,  and  test  its  teachings  by  his  experience,  and 
his  experience  by  its  teachings.  He  must  explore  the  vast 
domain  of  general  literature,  for  his  own  growth  and  that  of  his 
school.  He  must  be  faultless  in  the  subjects  he  is  to  teach,  as 
regards  his  technical  knowledge.  To  such  as  are  endowed  with 
industry  to  pursue  such  a  course,  this  book  is  sent  with  pleasure. 
In  their  hands  the  pupils  will  grow  to  honorable  manhood  and 
true  womanhood. 

State  Normal  School,  1  C.  M.   RITTER. 

Chico,  California, 

August  4th,  1891.  ) 


PART    I. 


THE  SUPPLY. 


"In  the  teaching  of  arithmetic  we  should  first  carry  the  pupils 
through  a  simple  and  comprehensive  course  of  calculation, 
embodying  all,  or  nearly  all,  the  fundamental  operations  of  num- 
bers, before  we  attempt  to  carry  them  through  the  so  called 
systematic  course  of  arithmetic,  involving  long  and  irksome 
calculations,  intended  to  give  expertness  and  skill  in  the  manip- 
ulation of  numbers,  rather  than  to  awaken  and  invigorate  the 
intellectual  faculties. ' '  —  Tate . 


ONE   TO   TEN.  13 


1    TO     1O. 


' '  Make  your  pupil  robust  and  healthy,  in  order  to  make  him 
reasonable  and  wise." — Rousseau. 


THE   BEGINNING    OF    NUMBER    WORK. 


The  experience  of  the  best  educators  of  the  world  seems  to 
point  to  substantially  the  same  method  of  developing  the  idea 
of  number.  Educators  differ  in  minor  matters  of  detail,  but  in 
the  general  plan  all  are  agreed.  This  is  because  the  nature  and 
workings  of  the  mind  have  been  studied  with  the  end  in  view  of 
determining  its  powers,  its  capabilities,  and  its  natural  inclina- 
tions when  in  a  normal  state.  The  result  of  these  studies  has 
been  the  definite  establishment  of  the  fact  that  in  experience,  in 
self-doing,  there  is  produced  the  greatest  increment  ol  education. 
Payne  says:  "Inasmuch  as  the  object  can  only  be  attained  by 
the  mental  action  of  the  learner,  by  his  observing,  remembering, 
etc.,  it  is  clear  that  what  he  does,  not  what  the  teacher  does,  is 
the  essential  part  of  the  process. "  It  is  therefore  conceded  that 
the  teacher's  chief  function  should  be  to  take  advantage  of  cir- 
cumstances, to  afford  means,  and  to  lead  and  direct,  so  that  the 
experience,  the  self-doing,  the  self-teaching,  may  be  as  effective 
as  possible.  "To  deny  this  principle  is  to  give  a  direct  sanction 
to  telling  and  cramming,  which  are  forbidden  by  the  laws  of 
education.  To  tell  the  child  what  he  can  learn  for  himself  is  to 
neutralize  his  efforts;  consequently,  to  defeat  all  the  ends  of  true 
education." 

The  child's  natural  methods  of  learning  are  objective;  there- 
fore objects  are  employed  as  the  representatives  of  ideas.  The 
child  naturally  learns  all  he  can  concerning  an  object;  therefore, 


14  THE  SUPPLY. 

we  infer  that  we  should  lead  him  to  exhaust  a  subject.  Hence, 
in  the  teaching  of  number,  objects  are  employed  until  the  con- 
cepts of  the  numbers  themselves  are  correct  and  ineradicable. 
The  numbers  are  considered  in  order,  and  everything  possible 
is  learned  concerning-  one  number  before  the  next  in  taken  up. 
Nothing  is  told  the  pupil  except  those  things  which  he  cannot  by 
any  possibility  learn  for  himself  under  the  skillful  direction  of  the 
teacher.  The  conversational  method  seems  particularly  adapted 
to  the  teaching  of  number.  By  it  the  pupils  are  unconsciously 
led  through  all  the  combinations  possible,  and  with  interest 
intensified  rather  than  abated.  "To  be  interesting,  the  questions 
must  deal  with  familiar  things,  must  be  varied,  and  must  be 
simply  expressed.  It  is  not  to  be  expected  that  a  lesson  of  this 
nature  can  succeed  unless  the  children  feel  that  the  teacher 
speaks  from  a  full  mind,  and  is  quite  at  ease." 

The  teacher  needs  to  be  ever  on  the  alert  for  weak  places. 
He  must  constantly  check  himself  in  his  impetuosity  to  go 
forward.  He  must  be  master  of  the  idea  that  to  be  over-desirous 
to  advance  his  pupils  is  as  dangerous  as  to  be  apathetic.  By 
means  of  numerous  and  varied  questions,  given  out  to  individuals 
and  to  the  class,  made  alive  by  the  use  of  objects  in  the  hands 
of  both  the  pupil  and  the  teacher,  and  by  means  of  reviews  and 
repetitions,  ever  repeated  yet  never  the  same,  the  patient  and 
tactful  teacher  will  seek  to  unfold  in  all  its  completeness  that 
most  inscrutable  of  creations,  the  human  mind.  Plato  says:  "If 
we  could  clearly  read  what  is  in  our  own  souls,  we  should  find 
there  a  correct  record  of  everything  proper  to  be  known. ' ' 

A  specimen  method  of  the  development  of  a  number  (the 
number  three,  for  instance)  is  given,  not  to  be  slavishly  adopted, 
but  to  be  studied  and  adapted  to  the  needs  and  surroundings  of 
the  particular  school.  If  the  scientific  principles  upon  which 
this  and  the  following  models  are  based  be  comprehended,  they 
will  be  of  much  vital  force  as  aids  to  a  systematic  and  thorough 
presentation  of  one  subject,  and  a  careful  preparation  for  its 
logically  connected  succeeding  subject.  It  is  hoped,  therefore, 
that  these  models  will  be  made  the  subjects  of  study,  and,  if  com- 
prehended and  approved,  that  they  will  be  adapted  to  the 
necessities  of  each  particular  teacher  and  class.  Many  times  it 
will  be  found  that  fewer  or  more  questions  will  be  necessary  to  a 
thorough  teaching  of  a  particular  point.  Different  teachers  will 


ONE   TO    TEN.  15 

differ  as  to  the  kind  or  the  number  of  objects,  the  kind  or  the 
number  of  questions  or  directions,  and  the  manner  of  presenting 
entirely  new  matter.  These  differences  are  quite  unimportant, 
provided  the  methods  of  each  teacher  be  based  upon  psychologi- 
cal principles  and  phenomena.  The  results  will  be  the  same. 
DeGarmo  says:  "Only  one  caution  needs  to  be  given.  The 
presupposition  of  brains  on  the  part  of  the  children  must  always 
be  made;  for  they  come  to  a  thousand  conclusions,  and  take  a 
thousand  steps,  in  thinking,  which  the  teacher  need  not  painfully 
point  out." 

The  numbers  one  and  two  need  to  be  treated  in  the  same 
logical  manner  and  with  the  same  degree  of  exactness  as  the 
number  three,  though  much  less  time  and  attention  will  be 
required  to  reach  perfect  results.  Time  however  is  not  the 
essence  of  the  subject.  Thoroughness  is  the  desideratum. 


THE  NUMBER  THREE. 


T.     (Holding  up  two  blocks.)     How  many  blocks  have  I? 
*C.     You  have  two  blocks. 

T.  (Holding  up  the  same  two  blocks  in  one  hand  and  picking 
up  another  block  with  the  other  hand.)  How  many  blocks 
have  I  now? 

C.     You  have  two  blocks  and  one  block. 

T.  (Putting  the  three  together.)  You  may  bring  me  just  as 
many  pencils. 

T.  You  have  brought  me  three  pencils.  You  may  now  bring 
me  three  marbles. 

T.     That  is  very  well  done;    now  how  many  blocks  have  I? 

C.     You  have  three  blocks. 

T.  (Holding  up  three  pencils  in  one  hand.)  How  many 
pencils  have  I? 

C.     You  have  three  pencils. 

T.     How  many  roses  have  I  ? 

C.     You  have  three  roses. 

T.     John,  you  may  bring  me  three  marbles. 

T.     Mary,  you  may  give  Susie  three  splints. 

*C.    Represents  "Child,"  "Children,"  or  "Class,"  according  to  circumstances. 


16  THE  SUPPLY. 

T.     Susie,  how  many  did  Mary  give  you  ? 

S.     She  gave  me  three. 

T.     How  many  marks  have  I  made  on  the  blackboard? 

C.     You  have  made  three  marks. 

T.  (Making  only  two  marks  this  time.)  How  many  marks 
have  I  made  this  time? 

C.     You  have  made  two  marks. 

T.  (Making  one  mark  a  little  at  the  right  of  the  two.)  How 
many  marks  have  I  made  this  time? 

C.     You  have  made  one  mark. 

T.     How  many  marks  did  I  make  both  times? 

C.     You  made  three  marks. 

T.     How  many  are  two  marks  and  one  mark? 

C.     Two  marks  and  one  mark  are  three  marks. 

T.  You  may  take  two  pencils  in  one  hand  and  one  pencil  in 
the  other  hand. 

T.     How  many  pencils  have  you  in  both  hands? 

C.     I  have  three  pencils  in  both  hands. 

T.     Two  pencils  and  one  pencil  are  how  many  pencils  ? 

C.     Two  pencils  and  one  pencil  are  three  pencils. 

T.  (Writing  upon  the  blackboard,  II  and  I  are  III.)  You 
may  read  what  I  have  written. 

C.     Two  and  one  are  three. 

T.  (Writing  upon  the  blackboard,  2  and  1  are  III.)  You 
may  read  what  I  have  written. 

C.     Two  and  one  are  three. 

T.     I  shall  now  show  you  another  way  to  make  three.     3. 

T.  (Writing  upon  the  blackboark,  2  and  1  are  3.)  You  may 
read  what  I  have  written. 

C.     Two  and  one  are  three. 

T.     (Holding  up  one  block.)     How  many  have  I  ? 

C.     You  have  one. 

T.  (Picking  up  two  with  the  other  hand.)  How  many  have 
I  in  this  hand? 

C.     You  have  two  in  that  hand. 

T.      How  many  have  I  in  both  hands? 

C.     You  have  three  in  both  hands. 

T.     One  and  two  are  how  many  ? 

C.     One  and  two  are  three. 

T.     You  may  bring  me  one  splint. 


ONE   TO   TEN.  17 

T.     You  may  now  bring  me  two  splints. 

T.     How  many  splints  did  you  bring  me  both  times  ? 

C.     I  brought  you  three  splints. 

T.     (Writing  1  and  2  are)     What  other  word  shall  I  write? 

C.     Three. 

T.     (Completing  the  sentence  with  3.)     You  may  read  it  now. 

C.     One  and  two  are  three. 

T.  (Placing  one  block  on  the  table.)  How  many  blocks  are 
on  the  table? 

C.     There  is  one  block  on  the  table. 

T.  (Picking  up  one  with  the  right  hand.)  How  many  have 
I  in  my  hand? 

C.     You  have  one  in  your  hand. 

T.  (Picking  up  another  with  the  other  hand.)  How  many 
have  I  in  this  hand? 

C.     You  have  one  in  that  hand. 

T.     How  many  are  there  on  the  table  and  in  my  two  hands  ? 

C.     Three. 

T.     One  and  one  and  one  are  how  many  ? 

C.     One  and  one  and  one  are  three. 

T.     You  may  bring  me  one  pencil. 

T.  (Holding  the  object  brought.)  You  may  now  bring  me 
one  splint. 

T.  (Holding  this  object  in  the  other  hand.)  You  may  now 
bring  me  one  tooth-pick. 

T.  (Having  all  three  dissimiliar  objects  in  full  view  of  the 
class.)  How  many  things  did  you  bring  me? 

C.     I  brought  you  three  things. 

T.     One  and  one  and  one  are  how  many  ? 

C.     One  and  one  and  one  are  three. 

T.  (Writing  1  and  1  and  1  are)  What  other  word  shall  I 
write  ? 

C.     Three. 

T.  (Completing  the  sentence  with  3.)  You  may  now  read 
what  I  have  written. 

C.     One  and  one  and  one  are  three. 

T.     Two  and  one  are  how  many? 

C.     Two  and  one  are  three. 

T.     One  and  two  are  how  many  ? 

C.     One  and  two  are  three. 


18  THE  SUPPLY. 

T.  One  and  one  and  one  are  how  many  ? 

C.  One  and  one  and  one  are  three. 

T.  (Holding  up  three  blocks.)     How  many  blocks  have  I? 

C.  You  have  three  blocks. 

T.  (Taking  one  away.)     How  many  have  I  taken  away? 

C.  You  have  taken  away  one. 

T.  How  many  are  left? 

C.  Two. 

T.  One  taken  away  from  three  leaves  how  many? 

C.  One  taken  away  from  three  leaves  two. 

T.  You  may  bring  me  three  marbles. 

T.  You  may  give  this  one  to  Hattie. 

T.  How  many  did  you  bring  me? 

C.  I  brought  you  three.  ] 

T.  How  many  did  you  give  to  Hattie  ? 

C.  I  gave  one  to  Hattie. 

T.  How  many  have  I  left? 

C.  You  have  two  left. 

T.  Three  less  one  are  how  many  ? 

C.  Three  less  one  are  two. 

T.  You  may  make  three  marks  on  the  blackboard. 

T.  You  may  now  erase  one. 

T.  How  many  are  there  now  ? 

C.  There  are  two. 

T.  How  many  did  you  erase? 

C.  One. 

T.  Then  three  less  one  are  how  many  ? 

C.  Three  less  one  are  two. 

T.  (Writing   3   less  1  are  2.)     You  may  read  what  I  have 
written. 

C.  Three  less  one  are  two. 

T.  (Holding  up  three  blocks.)     How  many  have  I? 

C.  You  have  three. 

T.  (Taking  two  away.)     How  many  have  I  taken  away? 

C.  You  have  taken  away  two. 

T.  How  many  have  I  left? 

C.  You  have  one  left. 

T.  You  may  place  three  blocks  on  your  desk. 

T.  You  may  now  put  two  of  them  in  your  desk. 

T.  How  many  have  you  left? 


ONE  TO  TEN.  19 

C.  I  have  one  left. 

T.  Then  three  blocks  less  two  blocks  are  how  many  blocks  ? 

C.  Three  blocks  less  two  blocks  are  one  block. 

T.  You  may  make  three  marks  on  your  slates. 

T.  You  may  erase  two  of  them. 

T.  How  many  are  left? 

C.  One. 

T.  Three  less  two  are  how  many  ? 

C.  Three  less  two  are  one. 

T.  (Writing  3  less  2  are)  You  may  tell  me  what  word  to 
write  last. 

C.  One. 

T.  (Writing  3  less  2  are  1.)  You  may  read  what  I  have 
written. 

C.  Three  less  two  are  one. 

T.  You  may  make  three  marks  on  the  blackboard. 

T.  You  may  erase  three  of  them. 

T.  How  many  are  left? 

C.  None. 

T.  You  may  bring  me  three  pieces  of  crayon. 

T.  I  shall  place  three  of  them  on  Mary's  desk. 

T.  How  many  have  I  left? 

C.  None. 

T.  Three  less  three  are  how  many  ? 

C.  Three  less  three  are  none. 

T.  (Writing  3  less  3  are)     What  word  shall    I    write  last? 

C.  Naught. 

T.  (Writing  3  less  3  are  0. )     You  may  read. 

C.  Three  less  three  are  naught. 

T.  Three  less  one  are  how  many  ? 

C.  Three  less  one  are  two. 

T.  Three  less  two  are  how  many? 

C.  Three  less  two  are  one. 

T.  Three  less  three  are  how  many  ? 

C.  Three  less  three  are  naught. 

T.  Place  one  pencil  on  the  table. 

T.  Place  another  one  with  it. 

T.  Place  another  one  with  them. 

T.  How  many  ones  have  you  placed  on  the  table  ? 

C.  Three  ones. 


20  THE  SUPPLY. 

T.  How  many  pencils  have  you  placed  there? 

C.  Three  pencils. 

T.  Three  ones  are  how  many? 

C.  Three  ones  are  three. 

T.  Here  are  some  pieces  of  crayon;  all  who  can  tell  me 
how  many  pieces  I  have  may  raise  their  right  hands. 

T.  How  many  are  there? 

C.  There  are  three. 

T.  Mary,  you  may  take  them  and  tell  us  how  many  ones  you 
find. 

M.  I  find  three  ones. 

T.  How  many  ones  are  there  in  three? 

C.  There  are  three  ones  in  three. 

T.  You  may  place  three  splints  in  a  pile. 

T.  You  may  now  see  how  many  twos  you  can  find. 

T.  How  many  twos  have  you,  Robert? 

R.  I  have  one  two. 

T.  Has  any  one  more  than  one  two  ? 

C.  I  have  one  splint  more  than  one  two. 

T.  How  many  have  one  two  and  one  more? 

C.  I  have. 

T.  Then  how  many  twos  are  there  in  three? 

C.  There  is  one  two  and  one  in  three. 

T.  Three  ones  are  how  many  ? 

C.  Three  ones  are  three. 

T.  How  many  ones  in  three? 

C.  There  are  three  ones  in  three. 

T.  How  many  twos  in  three? 

C.  There  is  one  two,  and  one  more,  in  three. 

T.  James  brought  me  two  roses  and  John  brought  me  one 

rose;  hdw  many  roses  did  both  bring  me? 

C.  Both  brought  you  three  roses. 

T.  Mary  has  three  apples  and  gives  one  apple  to  James;  how 
many  apples  has  Mary  left? 

C.  Mary  has  two  apples  left. 

T.  John  has  three  cents  and  spends  two  cents  for  candy;  how 
many  cents  has  he  left? 

C.  John  has  one  cent  left. 

T.  How  many  sticks  of  candy  at  one  cent  a  stick  can  you 
buy  for  three  cents? 


ONE  TO  TEN.  21 

C.  I  can  buy  three  sticks  of  candy. 

T.  Mary  has  one  book  and  Helen  has  two  books;  how  many 

books  have  both  ? 

C.  Both  have  three  books. 

T.  Delia  bought  three  oranges,  giving  one  cent  for  each 
orange;  how  many  cents  did  they  cost  her? 

C.  They  cost  her  three  cents. 

T.  Samuel's  mother  gave  him  three  cents  with  which  to  buy 
bananas  at  two  cents  a  piece;  how  many  bananas  did  he  buy? 

C.  He  bought  one  banana  and  had  one  cent  left. 

T.  Jessie  bought  three  peaches  and  gave  two  of  them  to  her 

sister;  how  many  had  she  left? 

C.  She  had  one  left. 

T.  James  has  a  cat,  a  dog,  and  a  rabbit;  how  many  pets 
has  he? 

C.  He  has  three  pets. 

T.  Two  and  how  many  are  three  ? 

C.  Two  and  one  are  three. 

T.  One  and  how  many  are  three? 

C.  One  and  two  are  three. 

T.  One  and  one  and  how  many  are  three  ? 

C.  One  and  one  and  one  are  three. 

T.  Three  less  how  many  are  two  ? 

C.  Three  less  one  are  two. 

T.  Three  less  how  many  are  one? 

C.  Three  less  two  are  one. 

T.  Three  less  how  many  are  naught? 

C.  Three  less  three  are  naught. 

T.  Three  ones  are  how  many  ? 

C.  Three  ones  are  three. 

T.  How  many  ones  in  three? 

C.  There  are  three  ones  in  three. 

T.  One  two  and  how  many  more  in  three? 

C.  There  is  one  two  and  one  more  in  three. 

T.  Read:  2  and  1  are  3. 

C.  Two  and  one  are  three. 

T.  This  is  also  written,  2  +  1  =  3. 

T.  Then  what  is  this:  +? 

C.  It  is  "and." 

T.  What  is  this:  =? 


±>  THE  SUPPLY. 

C.  It  is  "are." 

T.  You  may  now  read  this:  1  +  2=3. 

C.  One  and  two  are  three. 

T.  You  may  read :   1  + 1  +  1  =--  3. 

C.  One  and  one  and  one  are  three. 

T.  How  many  pints  in  a  quart  ? 

C.  There  are  two  pints  in  a  quart. 

T.  What  part  of  a  quart  is  one  pint  ? 

C.  One  pint  is  one-half  of  a  quart. 

T.  How  many  quarts  in  three  pints  ? 

C.  In  three  pints  are  one  quart  and  one  pint. 

T.  What  do  people  buy  by  the  quart  ? 

C.  They  buy  milk  by  the  quart. 

T.  (Holding  up  a  foot-rule.)  Who  can  tell  me  how  long 
this  is? 

Albert.     It  is  one  foot  long. 

T.  That  is  right;  this  is  one  foot. 

T.  (Holding  up  a  yard-stick.)  Who  can  tell  -what  I  have 
now? 

T.  It  is  a  yard. 

T.  What   is   it? 

C.  It  is  a  yard. 

T.  (Holding  up  the  foot-rule  again.)     What  is  this? 

C.  It  is   a  foot. 

T.  Now  count  how  many  feet  there  are  in  a  yard. 

T.  (Applying  the  foot  to  the  yard  three  times. )  How  many 
did  you  count? 

C.  I   counted   three. 

T.  Then  how  many  feet  in  a  yard  ? 

C.  There  are  three  feet  in  a  yard. 

T.  Who  can  tell  me  something  that  people  buy  by  the  yard  ? 

Florence.     They  buy  cloth  by  the  yard. 

T.  Yes;  people  buy  cloth  by  the  yard. 

T.  What  is  this? 

C.  It  is  a  foot. 

T.  What  is  this? 

C.  It  is  a  yard. 

T.  How  many  feet  in  a  yard? 

C.  There  are  three  feet  in  a  yard. 

T.  What  do  we  buy  by  the  yard? 


ONE  TO  TEN.  23 

C.     We  buy  cloth  by  the  yard. 

T.     You  may  show  me  the  pint,  William. 

T.     You  may  show  me  the  quart,  Ida. 

T.     How  many  pints  in  a  quart  ? 

C.     There  are  two  pints  in  a  quart. 

T.     One  pint  and  two  pints  are  how  many  quarts? 

C.     One  quart  and  one  pint. 

T.     At  one  cent  a  pint  what  will  one  quart  of  milk  cost  ? 

C.     One  quart  of  milk  will  cost  two  cents. 

T.     At  one  cent  a  foot  what  will  one  yard  of  ribbon  cost  ? 

C.     It  will  cost  three  cents. 

T.  John  has  three  cents  and  spends  two  cents  for  candy;  how 
many  cents  has  John  left? 

C.     John  has  one  cent  left. 

T.  James  brought  me  two  roses  and  John  brought  me  one 
rose;  how  many  roses  were  brought  me? 

C.     Three  roses  were  brought  you. 

T.  Here,  John,  is  a  three  cent  piece;  you  may  buy  candy 
with  it  at  one  cent  a  stick;  how  many  sticks  of  candy  can  you 
buy  with  the  three  cents? 

John.     I  can  buy  three  sticks  of  candy  with  the  three  cents. 

T.  Minnie,  I  shall  give  you  three  cents  and  you  may  buy 
oranges  at  two  cents  a  piece;  how  many  oranges  can  you  buy? 

Minnie.     I  can  buy  one  orange  and  have  one  cent  left. 

The  preceding  models  are  fairly  representative  of  the  character 
and  amount  of  work  necessary  in  teaching  the  respective  com- 
binations therein  developed.  The  same  general  plan  is  followed 
in  teaching  each  of  the  numbers  from  one  to  ten  inclusive.  Of 
course,  the  live  teacher  will  understand  that  the  plan  as  herein 
outlined  connot  be  strictly  adhered  to.  One  class  will  need 
more  and  another  class  may  require  less  drill  to  produce  a 
thorough  conception  of  the  subject.  The  questions  too  must  be 
addressed  to  the  members  of  the  class  individually,  or  to  the 
class  collectively,  in  accordance ,  with  circumstances;  no  set 
formula  can  be  mechanically  followed.  The  teacher  must  know 
human  nature  in  general,  and  that  of  his  class  in  particular; 
then,  with  wisdom  and  tact  such  as  the  successful  teacher  must 
needs  possess,  he  will  adapt,  rather  than  rigidly  adopt,  methods. 
The  purpose  of  adaptation  is  the  holding  of  interest.  Retain  the 


24  THE  SUPPLY. 

interest;  regain  the  interest;  or  change  the  subject.  It  is  better 
to  do  the  last  before  the  second  becomes  necessary. 

A  close  study  of  the  plan  outlined  discovers  that  it  is  purely 
inductive,  that  it  is  developmental,  and  that  it  is  exhaustive.  It 
will  also  be  seen  that  the  plan  incidentally,  yet  necessarily, 
comprehends  in  its  scope  a  most  thorough  and  continuous 
course  in  language.  All  answers  are  sought  to  be  made  in 
complete  sentences  that  are  precisely  responsive  to  the  questions 
asked.  This  is  thought  to  be  essentially  helpful  to  the  clear 
conception  of  the  number  idea,  and  therefore  aids  both  mathe- 
matically and  linguistically.  At  the  conclusion  of  the  develop- 
ment of  the  numbers  from  one  to  ten  inclusive,  all  the  combina- 
tions within  those  limits  should  be  made  by  the  pupils,  as  it 
were,  automatically;  that  is,  results  should  be  announced  without 
the  slightest  hesitation.  A  year,  of  ten  months,  is  conceded  to 
be  none  too  long  to  fulfill  these  demands.  Much  sooner,  some- 
times, it  seems  that  the  pupil  should  be  given  additional  lessons, 
but  as  the  subsequent  work  will  be  easy  or  difficult  in  accordance 
with  the  facility  with  which  the  pupils  announce  results  in  this 
elemental  period,  it  certainly  will  be  advantageous  to  ' '  make 
haste  slowly."  The  applied  work  can  be  varied  without  limit 
for  an  indefinite  period  of  time,  and  consequently  there  is  no 
danger  of  flagging  interest. 

The  work,  of  course,  is  almost  entirely  oral  during  the  first 
year;  yet  much  drill  of  the  following  nature  should  daily  be 
upon  the  blackboard,  and  should  be  read  and  completed  re- 
peatedly by  the  pupils  to  secure  familiarity  with  such  signs  and 
symbols  as  are  in  common  use: 


1+1= 

1—1  = 

1x1= 

1—1  = 

4-  of  2  = 

1+1+1= 

2-<l  = 

2x1  = 

2- 

-1  = 

iof  4  = 

2+1  = 

2—2  = 

3x1  = 

2- 

-2  = 

lof  6  = 

1+2  = 

3-1  = 

4x1  = 

3- 

_1  = 

iof  8  = 

1+1+1+1= 

3-2  = 

5x1  = 

3- 

-2= 

iof  10= 

2+1+1= 

3-3  = 

6x1  = 

3- 

-3  = 

iof  3  = 

1+2+1= 

4—1  = 

7x1  = 

4- 

-1  = 

Iof  6  = 

1+1+2= 

4—2  = 

8x1  = 

4- 

2  

iof  9  = 

2+2  = 

4-3  = 

9x1  = 

4- 

-3  = 

iof  4  = 

3+1  = 

4—4  = 

10x1= 

4- 

-4  = 

iof  8= 

1+3= 

5—1  = 

1x2= 

5- 

—  1  = 

iof  5  = 

1+1+1+1+1= 

5-2  = 

2x2= 

5- 

-2  = 

iof  10= 

ONE  TO  TEN. 


2+1+1+1= 

5—3  = 

3x2  = 

5-5-3: 

1+2+1+1= 

5-4  = 

4x2  = 

5H-4: 

1+1+2+1= 

5—5  = 

5x2= 

5-5-5 

1+1+1+2= 

6—1  = 

1x3= 

6-5-1 

2+2+1= 

6-2= 

2x3= 

6-5-2: 

2+1+2= 

6-3  = 

3x3= 

6-*-3: 

1+2+2  = 

6—4= 

1x4= 

6-5-4: 

3+1  +  1  = 

6-5  = 

2x4= 

6-5-5: 

1+3+1  = 

6-6  = 

1x5= 

6-5-6: 

1  +  1+3= 

7-1  = 

2x5= 

7-5-1: 

3+2= 

7—2  = 

1x6= 

7-^-2: 

2+3= 

7-3  = 

1x7  = 

7-^3: 

4  +  1  = 

7—4  = 

1x8= 

7-5-4: 

1+4  = 

7—5  = 

1x9= 

7-5-5: 

1+1+1+1+1+1= 

7-6= 

1x10= 

7-5-6: 

Etc. 

Etc. 

Etc. 

Also, 

1223345 

1121234 

3576 

233 

1121221 

-2  -4  -3  -2 

x2   x2   x3 

2)4  4) 

26  THE  SUPPLY. 


TENS,  1O   TO   1OO. 


"The  time,  end,  and  aim  of  all  our  work  is  the  harmonious 
growth  of  the  whole  being." — Froebel. 


After  the  numbers  from  one  to  ten  inclusive  have  been  thor- 
oughly learned,  then  the  tens  from  ten  to  one  hundred  inclusive 
should  be  taught  in  substantially  the  same  manner.  Much  less 
time  will  be  required,  however,  as  the  pupils'  minds  are  now 
receptive,  and  as  the  analogy  of  the  subjects  is  striking.  At  the 
conclusion  of  the  treatment  of  the  number  ten,  the  pupil  was 
prepared  for  the  new  series  by  having  the  ten  splints  carefully 
counted  and  tied  together  for  future  use.  The  bundles  should 
be  sufficiently  numerous  to  enable  each  pupil  to  have  ten  of 
them,  if  possible.  Ordinary  wooden  tooth-picks  will  be  found 
very  convenient,  and  quite  inexpensive,  for  this  purpose.  A 
variety  of  objects  is  no  longer  convenient  or  desirahle. 

T.     (Holding  up  one  bundle.)     How  many  have  I,  John? 

J.      You  have  ten. 

T.     You  may  hold  up  one  ten. 

T.  (Writing  10  on  the  blackboard. )  This  is  the  way  ten  is 
written. 

T.     You  may  hold  up  two  tens,  or  twenty. 

T.  (Writing  20  on  the  blackboard.)  This  is  the  way  two 
tens,  or  twenty,  is  written. 

T.     You  may  hold  up  three  tens,  or  thirty. 

T.  Who  will  write  three  tens,  or  thirty,  on  the  blackboard 
for  me? 

James.     I  will.     (Writes  30.) 

T.     That  is  right. 

T.     Clara,  you  may  show  me  four  tens,  or  forty. 


TENS,  TEN  TO  ONE  HUNDRED.  27 

T.  All  who  can  write  four  tens,  or  forty,  on  the  blackboard 

may  raise  their  hands. 

T.  Very  well;  Hazel,  you  may  write  four  tens,  or  forty,  on 
the  blackboard. 

H.  40. 

T.  (Writing  10,  20,  30,  40.)     You  may  read  the  first  number. 

C.  Ten. 

T.  The  second  number. 

C.  Two  tens,  or  twenty. 

T.  The  third  number. 

C.  Three  tens,  or  thirty. 

T.  The  fourth  number. 

C.  Four  tens,  or  forty. 

T.  Show  me  ten. 

T.  Show  me  twenty. 

T.  Show  me  ten  again. 

T.  Show  me  another  ten. 

T.  How  many  are  ten  and  ten  ? 

C.  Ten  and  ten  are  twenty. 

T.  Ella,  you  may  write,  Ten  and  ten  are  twenty,  on  the 
blackboard. 

E.  10  +  10=20. 

T.  Hold  up  twenty. 

T.  Take  away  ten. 

T.  How  many  remain? 

C.  Ten  remain. 

T.  Twenty  less  ten  are  how  many  ? 

C.  Twenty  less  ten  are  ten. 

T.  Maggie,  you  may  write,  Twenty  less  ten  are  ten. 

M.  20—10=10. 

T.  Holding  up  twenty  again. 

T.  How  many  tens  are  you  holding  up? 

C.  Two  tens. 

T.  How  many  tens  are  twenty? 

C.  Two  tens  are  twenty. 

T.  William,  you  may  write,  Two  tens  are  twenty. 

W.  2x10=20. 

T.  Show  me  ten. 

T.  Show  me  another  ten. 

T.  Show  me  another  ten. 


28  THE  SUPPL  Y. 

T.  How  many  have  you  shown  me? 

C.  I  have  shown  you  thirty. 

T.  Then  ten  and  ten  and  ten  are  how  many  ? 

C.  Ten  and  ten  and  ten  are  thirty. 

T.  Charles,  you  may  write  it  upon  the  blackboard. 

C.  10+10+10-30. 

T.  How  many  tens  are  there  in  thirty? 

C.  There  are  three  tens  in  thirty. 

T.  Show  me  twenty. 

T.  Show   me   ten  more. 

T.  How  many  are  twenty  and  ten  ? 

C.  Twenty  and  ten  are  thirty. 

T.  Sarah,  you  may  write  it. 

S.  20+10-30. 

T.  Place  thirty  upon  your  desk. 

T.  Take  ten  of  them  away. 

T.  How  many  remain? 

C.  Twenty  remain. 

T.  Thirty  less  ten  are  how  many? 

C.  Thirty  less  ten  are  twenty. 

T.  Frank  may  write  this  one. 

F.  30—10=20. 

T.  Take  away  another  ten. 

T.  How  many  have  you  taken  away  in  all  ? 

C.  I  have  taken  away  twenty. 

T.  How  many  remain? 

C.  Ten  remain. 

T.  Thirty  less  twenty  are  how  many? 

C.  Thirty  less  twenty  are  ten. 

T.  Beatrice  may  write. 

B.  30-20-10. 

T.  Take  away  another  ten. 

T.  How  many  did  you  have  at  first? 

C.  I  had  thirty  at  first. 

T.  How  many  have  you  taken  away? 

C.  I  have  taken  away  three  tens,  or  thirty. 

T.  Then  thirty  less  thirty  are  how  many  ? 

C.  Thirty  less  thirty  are  naught. 

T.  Kate  may  write. 

K.  30-30-0. 


TENS,  TEN  TO  ONE  HUNDRED.  29 

T.  Three  tens  are  how  many? 

C.  Three  tens  are  thirty. 

T.  Vesta  may  write. 

V.  3x10=30. 

T.  How  many  tens  in  thirty  ? 

C.  There  are  three  tens  in  thirty. 

T.  See  how  many  twenties  you  can  find  in  thirty. 

C.  There  are  one  twenty  and  ten  in  thirty. 

T.  (Holding  up  four  bundles.)     How  many  have  I? 

C.  You   have   forty. 

T.  (Taking  away  ten.)     How  many  have  I  left? 

C.  You  have  thirty  left. 

T.  Forty  less  ten  are  how  many? 

C.  Forty  less  ten  are  thirty. 

T.  Olive,  you  may  write. 

O.  40—10=30. 

T.  (Taking  away   another   ten.)     How  many  have  I  taken 

away  both   times  ? 

C.  You  have  taken  away  two  tens,  or  twenty. 

T.  Forty  less  twenty  are  how  many? 

C.  Forty  less  twenty  are  twenty. 

T.  Lorain  mav  write. 

L.  40—20=20. 

T.  How  many  did  I  take  away  ? 

C.  You  took  away  twenty. 

T.  How  many  have  I  ? 

C.  You  have  twenty. 

T.  Then  how  many  twenties  are  there  in  forty  ? 

C.  There  are  two  twenties  in  forty. 

T.  Maud  may  write. 

M.  2x20=40. 

T.  You  may  show  us  forty,  Stella. 

T.  Show  us  how  many  thirties  you  can  find. 

T.  How  many  thirties  in  forty? 

S.  One  thirty  and  ten  remainder. 

T.  How  many  tens  in  forty? 

C.  There  are  four  tens  in  forty. 

T.  Ella  may  write. 

E.  4x10=40. 

T.  How  many  tens  in  ten? 


30  THE  SUPPLY. 

C.  There  is  one  ten  in  ten. 

T.  Two  tens  are  how  many  ? 

C.  Two  tens  are  twenty. 

T.  Three  tens  are  how  many  ? 

C.  Three  tens  are  thirty. 

T.  How  many  are  two  twenties? 

C.  Two  twenties  are  forty. 

T.  You  may   all  copy  these  on  your  slates   or   papers   and 
complete  them: 

30  +  10=  2x20= 

10+10=  1x10  = 

3x10=  10^-10  = 

20-=-10=  30—30= 

10+10+10=  20  +  20= 

30—20=  40—20= 

2x10=  40-^10= 

T.  You  may  place  forty  on  your  desks. 

T.  How  many  tens  are  there  in  forty? 

C.  There  are  four  tens  in  forty. 

T.  You  may  now  place  another  ten  with  them. 

T.  How  many  tens  have  you  now  ? 

C.  I  have  five  tens. 

T.  Anna,  you  may  write  the  number  for  five  tens,  or  fifty. 

A.  50. 

T.  How  many  are  forty  and  ten  ? 

C.  Forty  and  ten  are  fifty. 

T.  Robert  may  place  it  on  the  board. 

R.  40+10=50. 

T.  How  many  are  ten  and  forty? 

C.  Ten  and  forty  are  fifty. 

T.  Harry  may  write. 

H.  10+40=50. 

T.  Hold  up  thirty  in  one  hand. 

T.  Hold  up  twenty  in  the  other  hand. 

T.  How  many  have  you  in  both  hands  ? 

C.  I  have  fifty  in  both  hands. 

T.  How  many  are  thirty  and  twenty  ? 

C.  Thirty  and  twenty  are  fifty. 

T.  Dora  may  write. 


TENS,  TEN  TO  ONE  HUNDRED,  31 

D.  30+20=50. 

T.  Twenty  and  thirty  are  how  many? 

C.  Twenty  and  thirty  are  fifty. 

T.  Esther  may  write  for  us  a  little  while. 

E.  20+30-50. 

T.  Fifty  less  ten  are  how  many  ? 

C.  Fifty  less  ten  are  forty. 

E.  50-10=40. 

T.  Fifty  less  twenty  are  how  many? 

C.  Fifty  less  twenty  are  thirty. 

E.  50—20=30. 

T.  Fifty  less  thirty  ? 

C.  Fifty  less  thirty  are  twenty. 

E.  50-30=20. 

T.  Fifty  less  forty? 

C.  Fifty  less  forty  are  ten. 

E.  50—40=10. 

T.  Fifty  less  fifty? 

C.  Fifty  less  fifty  are  naught. 

E.  50—50=0. 

T.  How  many  tens  in  fifty? 

C.  There  are  five  tens  in  fifty. 

E.  5x10=50. 

T.  See  how  many  twenties  you  can  find  in  fifty. 

C.  There  are  two  twenties  and  a  ten  in  fifty. 

E.  2x20  +  10=50. 

T.  How   many   thirties? 

C.  One   thirty   and  a   twenty   in   fifty. 

E.  1x30  +  20=50. 

T.  You  may  place  six  tens,  or  sixty,  together. 

E.  60. 

T.  Esther   is   excused   and   Mary   may   write. 

T.  Place   seven   tens,    or   seventy,    together. 

M.  70. 

T.  Eight   tens,    or   eighty. 

M.  80. 

T.  Nine   tens,   or   ninety. 

M.  90. 

T.  Ten   tens,    or   6ne   hundred. 

M.  100. 


32  THE  SUPPLY. 

T.  How   many   tens   in  sixty? 

C.  There   are   six   tens    in   sixty. 

M.  6x10=60. 

T.  How   many   tens   in   seventy? 

C.  There   are   seven   tens   in  seventy. 

M.  7x10=70. 

T.  How   many   tens   in   eighty? 

C.  There  are   eight   tens   in  eighty. 

M.  8x10=80. 

T.  How   many   tens   are   ninety  ? 

C.  Nine   tens   are    ninety. 

M.  9x10=90. 

T.  How   many   tens   are   one   hundred? 

C.  Ten   tens   are   one   hundred. 

M.  10x10=100. 

T.  Mary  is   excused. 

T.  How   many   are   eight   tens? 

C.  Eight   tens  are   eighty. 

T.  Seven  tens? 

C.  Seven   tens   are   seventy. 

T.  Nine   tens  ? 

C.  Nine   tens   are   ninety. 

T.  Six   tens? 

C.  Six  tens  are  sixty. 

T.  Ten  •  tens  ? 

C.  Ten   tens  are   one   hundred. 

T.  Fifty   and   ten? 

C.  Fifty   and   ten   are   sixty. 

T.  Ida  will   please   write   for    us. 

I.  50+10=60. 

T.  Sixty  and   ten? 

C.  Sixty  and   ten   are   seventy. 

I.  60+10=70. 

T.  Ninety   and   ten? 

C.  Ninety   and   ten   are   one   hundred. 

I.  90+10=100. 

T.  Fifty   and  twenty? 

C.  Fifty   and   twenty   are  seventy. 

I.  50+20=70. 

T.  Seventy   and   twenty? 


TENS,  TEN  TO  ONE  HUNDRED.  33 

C.  Seventy   and   twenty   are   ninety. 

I.  70+20=90. 

T.  Eighty   and  twenty? 

C.  Eighty   and   twenty   are  one   hundred. 

I.  80+20-100. 

T.  Sixty   and   thirty? 

C.  Sixty   and   thirty  are    ninety. 

I.  60+30=90. 

T.  Forty   and   thirty? 

C.  Forty   and   thirty   are   seventy. 

I.  40+30=70. 

T.  Seventy   and  thirty? 

C.  Seventy   and   thirty  are   one   hundred. 

I.  70+30=100. 

T.  Forty   and   forty? 

C.  Forty   and   forty   are   eighty. 

I.  40+40=80. 

T.  How   many   forties   are  eighty? 

C.  Two   forties   are   eighty. 

I.  2x40=80. 

T.  Fifty   and   fifty? 

C.  Fifty   and   fifty   are   one   hundred. 

I.  50+50=100. 

T.  How   many   fifties   in   one  hundred? 

C.  There   are   two   fifties   in   one  hundred. 

I.  2x50=100. 

T.  See   how  many   thirties   are   in  ninety. 

•C.  There   are   three   thirties   in   ninety. 

I.  3x30=90. 

T.  Walter,    will   you   please   relieve   Ida? 

T.  See   how   many   twenties   there   are   in   one   hundred. 

C.  There   are  five   twenties   in   one   hundred. 

W.  5x20=100. 

T.  Twenties   in   eighty? 

C.  Four   twenties   in   eighty. 

W.  4x20=80. 

T.  Place   twenty   on   your  desk. 

T.  Bring   me   one-half  of  them. 

T.  How   many   did   you   bring   me? 

C.  I    brought   you   ten. 


34 


THE  SUPPLY. 


T.     Then   what   is   one-half  of  twenty? 

C.     One-half  of  twenty    is   ten. 

T.     Place   forty   on   your  desk. 

T.     Bring   me   half  of  them. 

T.     One-half  of  forty   is   how   many? 

C.     One- half  of  forty   is   twenty. 

T.     Place   sixty  on   your   desk. 

T.     Divide   them    into   two   piles  just  alike. 

T.     Bring   me   one-half  of  them. 

T.     What   is   one-half  of  sixty? 

C.     One-half  of  sixty   is   thirty. 

T.     What   is   one-half  of  eighty? 

C.     One-half  of  eighty    is   forty. 

T.     What   is   one-half  of  one   hundred? 

C.     One-half  of  one  hundred    is   fifty. 

T.     You   may   copy   and   finish: 
40+10=  10x10  = 

60—10=  8x10  = 

70+20=  80—70  = 

2x30=  100-50= 

100—10=  50+50= 

80+20=  i  of20= 

90-40=  |  of  60= 

2x50=  70—50= 

|  of  100=  60—50= 

2x40=  30-4-10= 

\  of  80=  60-4-20= 

60—20=  90n-30= 

70—40=  100-4-50= 

90-60=  20+30+10+40= 

4x20=  80—60= 

30+60+10=         60—30= 

40+40+20=  5x20= 


1x1  = 

1x2  = 

1x10= 

1x20= 

2x1  = 

2x2= 

2x10= 

2x20  = 

3x1  = 

3x2  = 

3x10= 

3x20= 

4x1  = 

4x2  = 

4x10  = 

4x20  = 

5x1  = 

5x2= 

5x10= 

5x20= 

6x1  = 

1x3  = 

6x10  = 

1x30= 

7x1  = 

2x3  = 

7x10  = 

2x30= 

8x1  = 

3x3  = 

8x10= 

3x30= 

9x1  = 

1x4  = 

9x10  = 

1x40= 

10x1  = 

2x4  = 

10x10= 

2x40= 

1x5= 

1x50= 

2x5  = 

2x50  = 

TENS,  TEN  TO  ONE  HUNDRED. 


35 


This  work  should  be  continued  until  the  pupils  are  perfectly 
familiar  with  the  names  and  the  symbols  of  the  tens,  and  are 
thoroughly  aware  of  the  fact  that  units  and  tens  combine  pre- 
cisely alike  and  that  they  differ  only  in  the  tens  being  groups 
instead  of  single  units,  in  having  different  names,  and  in  having 
the  tens  stand  one  place  to  the  left  with  a  naught  at  the  right. 
At  this  stage  of  the  pupils'  advancement,  familiarity  with  the 
subsidiary  coins,  their  appearance,  value,  and  equivalence,  should 
be  made  thorough,  since  all  except  the  twenty-five  cent  piece 
may  be  expressed  with  numbers  with  which  they  are  conversant. 
Matters  of  making  change  should  be  especially  emphasized 
and  made  clear. 


36  THE  SUPPLY. 


1O    TO    2O. 


"Observe  the  nature  and  propensities  of  your  children,  in 
order  to  be  able  to  educate  them  according  to  their  individual 
wants  and  talents," — Pestalozzi. 


The  chief  difficulty  in  number  work,  after  the  units  and  tens 
are  separately  comprehended,  consists  in  giving  a  clear  under- 
standing of  the  two  combined  from  ten  to  twenty;  for,  if  this 
step  be  thoroughly  taken,  the  combinations  from  twenty  to 
thirty,  thirty  to  forty,  etc.,  will  present  no  serious  impediments 
to  the  progress  of  the  student.  Hence  this  step  should  be 
taken  with  much  deliberation. 

By  means  of  a  bundle  of  ten  and  a  single  splint,  develop  a 
clear  conception  of  ten  and  one,  or  eleven,  and  the  manner  of 
expressing  it,  asking  the  pupil  to  tell  you  which  one  of  the 
ones  represents  the  bundle,  or  ten,  and  which  one  represents 
the  single  splint. 


ELEVEN. 


T.  Show  me  ten. 

T.  Show   me   one. 

T.  Place   the   ten   and   the  one  together. 

T.  You  now  have  eleven.     Eleven  is   written   thus:    11, 

T.  How   many   did   you   show   me? 

C.  I   showed   you   eleven. 

T.  Show   me   another   eleven. 

T.  (Holding   up  eleven.)     How   many   have  I? 

C,  You   have   eleven. 


TEX  TO  TWENTY.  37 

T.  Show   me   which   of  the  ones   is   the  ten. 

T.  Show   me   which   of  the   ones   is  the   one. 

T.  (Erasing   the  eleven.)     Mettie .  may   write   eleven. 

M.  11. 

T.  You   may   all   place   eleven   on   your  desks. 

T.  Look  at  them  and  tell   me  how  many  ten  and  one  are. 

C.  Ten  and   one   are   eleven. 

T.  You   may  all   write  that  on   your  slates. 

C.  10+1=11. 

T.  Eleven   less   one   are  how  many? 

C.  Eleven   less    one  are   ten.     (11 — 1=10.) 

T.  Eleven   less   ten   are   how   many? 

C.  Eleven   less   ten  are  one.     (11 — 10=1.) 

T.  You  may  now  remove  the  elastic  from  the  bundle,  but 

leave  the  one  and  the  bundle  apart. 

T.  You  may  now  take  one  from  the  bundle  and  place  it 

with  the  one. 

T.  How  many   are  ten   less   one? 

C.  Ten   less  one   are   nine. 

T.  Then   how   many   does   that   leave   in   the  large  pile? 

C.  It  leaves  nine. 

T.  How  many  are   in   the   small  pile? 

C.  Two   are  in   the   small  pile. 

T.  How  many  are  in  both  piles? 

C.  Eleven   are  in  both  piles. 

T.  Then   how   many  are   nine  and   two? 

C.  Nine  and   two   are  eleven.     (9+2=11.) 

T.  How   many  are   two  and   nine? 

C.  Two   and   nine  are   eleven.     (2+9=11.) 

T.  Place   your   hand   over  the   two. 

T.  How   many   are  not  covered? 

C.  Nine   are   not   covered. 

T.  Then   eleven   less   two  are   how   many? 

C.  Eleven   less  two   are   nine.     (11 — 2=9.) 

T.  Eleven   less   nine  are   how  many? 

G.  Eleven   less  nine   are   two.     (11 — 9=2.) 

T.  You  may  now  take  another  one  from  the  large  pile  and 

place  it  with  the  small  pile. 

T.  How  many  are  left  in   the   large   pile? 

C.  Eight  are  left  in  the  large  pile. 


61233 


38  THE  SUPPLY. 

T.  How   many   are   in  the  small  pile? 

C.  Three  are  in  the  small   pile. 

T.  Then  eight   and   three   are  how   many? 

C.  Eight   and   three   are   eleven.      (8+3=11.) 

T.  Three  and   eight   are   how   many? 

C.  Three   and   eight   are   eleven.     (3+8—11.) 

T.  Place   your  hand   over   the   three. 

.T.  Eleven   less  three   are   how   many? 

C.  Eleven  less  three  are  eight.     (11 — 3=8.) 

T.  Place  your  hand  over  the  eight. 

T.  Eleven   less  eight  are  how  many? 

C.  Eleven  less  eight  are  three.     (11—8=3.) 

T.  Take   one  from  the   eight   and   place   it   with  the  three. 

T.  How  many  are  now  in  the  different  piles? 

C.  Seven  in  the  one  and  four  in  the  other. 

T.  Then  seven  and  four  are  how  many? 

C.  Seven  and  four  are  eleven.     (7+4=11.) 

T.  Four  and  seven  are  how  many? 

C.  Four  and  seven  are  eleven.     (4+7=11.) 

T.  Eleven  less  four  are  how  many? 

C.  Eleven  less   four  are  seven.     (11 — 4  =  7.) 

T.  Eleven  less  seven  are  how  many  ? 

C.  Eleven  less  seven  are  four.     (11 — 7=4.) 

T.  Take  another  from  the  larger  pile  and  place  it  with  the 
smaller. 

T.  How  many  are  now  in  each  pile? 

C.  Six  in  the  one  and  five  in  the  other. 

T.  Six  and  five  are  how  many? 

C.  Six  and  five  are  eleven.     (6+5=11.) 

T.  Five  and  six  are  how  many? 

C.  Five  and  six  are  eleven.     (5+6 =1L) 

T.  Eleven  less  five  are  how  many? 

C.  Eleven  less  five  are  six.     (11 — 5=6.) 

T.  Eleven  less  six  are  how   many? 

C.  Eleven  less  six  are  five.     (11 — 6=5.) 

The  teacher  then  writes  the  following  upon  the  blackboard 
in  regular  order  and  calls  upon  the  class  collectively,  and  then 
individually,  to  read,  and  to  supply  the  correct  result.  This 
exercise  needs  to  be  carried  on  with  the  eleven  upon  their 

desks,  so  that  in  every  case  of  doubt  or  error  the  pupils  may 


TEN  TO  TWENTY.  39 

objectively  reach   the  correct  conclusion.     Plenty   of  time   and 
plenty  of  patience  need  to  be  employed  in  this  work: 

10+1=  10+  -11.  +1  =  11. 

1+10=  1+  -11.  +10  =  11. 

11—1=  11-  =10.  -1  =  10. 

11-10=  11-  =1.  -10  =  1. 

9+2=  9+  =11.  +2  =  11. 

2+9=  2+  =11.  +9  =  11. 

11—2=  11-  =9.  —2  =  9. 

11—9=  11-  =2.  —9  =  2. 

8+3=  8+  =11.  +3  =  11. 

3+8=  3+  =11.  +8  =  11. 

11—3=  11-  -8.  —3  =  8. 

11—8=  11-  =3.  —8  =  3. 

7+4  =  7  +  =11.  +4  =  11. 

4+7=  4  +  =11.  +7  =  11. 

11—4=  11-  =7.  -4  =  7. 

11—7=  11-  =4.  -7  =  4. 

6+5=  6+  =11.  +5  =  11. 

5+6=  5+  =11.  +6  =  11. 

11—5=  11-  =6.  —5  =  6. 

11—6=  11-  =5.  —6=5. 

T.     See  how  many  twos  there  are  in  eleven. 

C.     There    are    five     twos    and    one    remainder    in    eleven. 
(5x2+1=11.) 

T.     See  how  many  threes  there  are. 

C.     There  are  three  threes  and  two  remainder.     (3x3+2  = 
11.) 

T.     How  many  fours  are  there? 

C.     There   are   two  fours  and   three  remainder.     (2x4+3= 
11.) 

T.     How  many  fives  are  there? 

C.     There  are  two  fives  and  one  remainder.     (2x5+1  =  11.) 

The   teacher   then   should  give  blackboard  and  seat  work  as 
follows : 

5x2+1=  12312469 

3x3+2=  35221121 

2x4+3^  _Z_?_2    J*  J>  _3  _2  J; 

2x5+1= 


40  THE  SUPPL  Y. 

2+3+5+1= 

4+2+2+3= 

7+4—6=  11       11       11       11       11       11         11       11 

8  +  3-2=:  -4     —3     -^6     -7    —5     —9     -10     -1. 

6+5—7  = 

8+3-5  = 

In  like  manner  place  two  splints  with  the  bundle,  and  ask 
the  pupil  to  write  on  the  board  what  you  have  shown  him. 
He  will  doubtless  write  12  quickly,  since  he  knows  where  to 
write  the  10  and  where  the  2;  it  then  only  remains  to  give  the 
number  a  name.  Then  develop  the  number  fully,  in  accord- 
ance with  the  plan  followed  with  the  eleven.  This  plan  should 
be  continued  in  the  same  general  manner  with  the  other  num- 
bers to  twenty;  after  which  the  numbers  should  be  counted 
from  one  to  twenty,  forward  and  backward,  concretely  and 
abstractly,  by  ones,  by  twos,  by  threes,  by  fours,  and  by  fives, 
beginning  for  the  twos  at  one  and  two  indifferently;  for  the 
threes  at  one,  two,  and  three,  indifferently;  for  the  fours  at 
one,  two,  three,  and  four,  indifferently;  and  for  the  fives  at 
one,  two,  three,  four,  and  five,  indifferently.  Such  combinations 
of  two  numbers  less  than  ten  as  will  produce  a  number  be- 
tween ten  and  twenty,  (as  for  instance,  six  and  five,  eight  and 
four,  seven  and  five,  nine  and  four,  seven  and  six,  nine  and 
eight)  require  to  be  made  so  familiar  that  the  answer  becomes 
merely  a  mechanical  vocalization  of  the  concept  drawn  from 
the  visible  or  audible  percept.  The  mode  of  reaching  this  result 
may  be  by  tabulations,  as  follows,  which  should  be  made  by 
the  pupils: 

11=1  +  10.      12=2  +  10.      13=3  +  10.     14=4+10. 
11=2+9.        12=3+9.        13=4+9.       14=5+9. 
11=3  +  8.        12=4+8.        13=5+8.       14=6+8. 
11=4+7.        12=5  +  7.        13=6+7.       14  =  7  +  7. 
11=5+6.        12=6+6. 

15=5+10.      16=6+10.      17  =  7  +  10.     18=8+10.     19  =  9+10. 
15=6+9.        16  =  7+9.        17=8+9.       18=9  +  9. 
15  =  7  +  8.        16  =  8+8. 

From  the  foregoing,  the  pupils  may  make  subtraction  tables 
as   follows : 
11—1  =  10.  12—2=10.  13—3=10.  14—4=10. 


TEN  TO  TWENTY. 


41 


11-2=0. 

12-3=9.                13-4=9. 

14—5=9. 

11-3=8. 

12—4=8.                13—5=8. 

14—6=8. 

11—4=7. 

12-5  =  7.                13—6=7. 

14-7  =  7. 

11—5=6. 

12—6=6.                13—7=6. 

14—8=6. 

11—6=5. 

12—7=5.                13-8=5. 

14—9=5. 

11-7=4. 

12-8=4.                13—9=4. 

14-10=4. 

11—8=3. 

12—9=3.                13—10=3. 

11—9=2. 

12—10=2. 

11-10=1. 

15—5=10. 

16—6  =  10.      17—7  =  10.     18—8  =  10. 

19—9  =  10. 

15-6=9. 

16-7=9.        17—8=9.       18-9=9. 

19—10=9. 

15—7=8. 

16—8=8.        17—9=8.,      18—10=8. 

15-8=7. 

16—9  =  7.        17—10=7. 

15-9=6. 

16-10=6. 

15—10=5. 

Then  such 

combinations  as  the  following   may 

be   tabulated 

by  the   pupils: 

12=2x6. 

14=2x7.         15=3x5.       16=2x8. 

18=2x9. 

12=3x4. 

14  =  7x2.         15=5x3.       16=4x4 

18=3x6. 

12=4x3. 

16=8x2. 

18=6x3. 

12=6x2. 

18=9x2. 

20=2x10. 

i 

20=4x5. 

20=5x4. 

20  =  10x2. 

12-5-2=6. 

14-2  =  7.         15^-3=5.       16-2=8. 

18-5-2=9. 

12^-3=4. 

14-i-7=2.         15-f-5=3.       16-!-4=4. 

18-^3=6. 

12-s-4=3. 

16-i-8=2. 

18-s-6=3. 

12-5-6  =  2. 

18-^9=2. 

20n-2  =  10. 

20-i-4=5. 

20-f-5=4. 

20-s-10=2. 

11=2x5+1. 

13=2x6+1.        17=2x8+1. 

19=2x9+1. 

11=3x3+2. 

13=3x4  +  1.        17=3x5+2. 

19=3x6+1. 

11=4x2+3. 

13=4x3  +  1.        17=4x4  +  1. 

19=4x4+3. 

42  THE  SUPPLY. 

11=5x2  +  1. 


13-5x2+3. 
13=6x2  +  1. 

17=5x3  +  2. 
17=6X2  +  5. 

17  =  7x2  +  3. 
17=8x2  +  1. 

19=5x3+4. 
19=6x3  +  1. 
19  =  7x2  +  5. 

19=8x2  +  3. 
19=9x2  +  1. 

The  following  indicates  the  character  of  the  work  that  should 
also  receive  much  attention  at  this  time: 

157581  10  10          30  20 

333294          20          10          40  10 

420616          30          20  10          30 

7          9          8          3          2          3          40          30  10  20 

9  9  10  11  12  14 

—5  —3  -4  -7  —5  —6 

Applications  like  the  following  should  each  day  be  given  the 
pupils,  illustrating  with  the  measures  themselves,  so  that  as  the 
work  goes  on  the  tables  of  compound  numbers  shall  be  uncon- 
sciously learned: 

There  are  two  pints  in  one  quart;  how  many  pints  are  there 
in  two  quarts? 

There  are  seven  days  in  one  week;  how  many  days  are  there 
in  two  weeks? 

How  many  cents  in  two  five-cent  pieces? 

How  many  cents  in  three  three-cent  pieces? 

How  many  cents  in  one  dime? 

In  one  month  there  are  four  weeks;  how  many  weeks  in 
two  month  ? 

With  five  cents  I  bought  two  oranges  at  two  cents  a  piece: 
how  many  cents  did  I  have  left? 

In  one  gallon  are  four  quarts;  how  many  quarts  in  two 
gallons? 

In  one  dime  are  ten  cents;  how  many  cents  in  three  dimes? 


TWENTY  TO  ONE  HUNDRED.  43 


2O     TO     1OO. 


"As  a  rule  the  poorest  teachers  talk  most." — Sheib. 


In  order  that  the  concepts  of  the  numbers  between  twenty 
and  thirty,  thirty  and  forty,  forty  and  fifty,  etc.,  shall  be  per- 
fect, the  bundles  of  ten  each  should  be  associated  with  the 
single  splints  until  a  few  numbers,  as  twenty-one,  twenty-two, 
twenty-three,  and  twenty-four,  are  taught,  when  the  same 
method  may  be  employed  for  thirty-one,  forty-one,  and  fifty- 
one,  after  which  the  objects  may  be  wholly  dropped,  except 
in  the  case  of'  a  dull  or  confused  pupil  for  whom  it  may  be 
necessary  to  return  to  the  objects  occasionally.  The  numbers 
from  twenty  to  one  hundred  may  be,  and  in  fact  should  be, 
taught  simultaneously.  After  a  clear  idea  of  the  units  and 
tens  that  make  up  the  numbers  is  acquired,  counting  by  tens 
from  ten  to  one  hundred  should  be  reviewed,  so  that  while 
counting  otherwise  no  omissions  or  reduplications  shall  occur. 
Counting  by  ones  to  one  hundred  forward  and  backward,  then 
by  fives  in  Mike  manner,  then  by  elevens  in  like  manner,  then 
by  twos  in  like  manner,  should  be  added.  These  are  all  very 
simple  combinations  that  will  not  tax  the  minds  of  the  pupils 
severely  at  all.  This  work  should  be  followed  by  tabulations 
as  follows : 

1  +  2=3.  2  +  2-4.  3  +  2-5.  4  +  2=6. 

11  +  2  =  13.  12  +  2  =  14.  13+2  =  15.  14  +  2  =  16. 

21  +  2  =  23.  22  +  2  =  24.  23  +  2  =  25.  24  +  2=26. 

31  +  2=33.  32  +  2=34.  33  +  2  =  35.  34  +  2=36. 

Etc.  to  Etc.  to  Etc.  to  Etc.  to 

91  +  2=93.  92  +  2=94.  93+2=95.  94  +  2=96. 


44  THE  SUPPLY. 

5+2  =  7.         0  +  2  =  8.          7  +  2=9.         8  +  2  =  10.  9+2=11. 

Etc.  to           Etc.  to            Etc.  to             Etc.  to  Etc.  to 

95+2=97.  96  +  2=98.     97+2=99.     98+2=100.  89+2=91. 


1+3=4. 

2+3=5.                 3+3=6. 

4+3=7. 

Etc.  to 

Etc  to                     Etc.  to 

Etc.  to 

91+3=94. 

92+3=95.              93+3=96. 

94+3=97. 

5+3=8. 

6+3=9.          7  +  3  =  10.           8+3  =  11. 

9+3  =  12. 

Etc  to. 

Etc.  to             Etc.  to                Etc.  to 

Etc.  to 

95+3=98. 

96+3  =  99.     97+3  =  100.     88+3=91. 

89+3=92. 

Etc.  to 

1  +  9  =  10. 

2+9  =  11.               3  +  9=42. 

4+9=13 

Etc.  to 

Etc.  to                    Etc.  to 

Etc.  to 

91+9=100. 

82+9=91.             83+9=92. 

84+9=93. 

5  +  9=14. 

6+9  =  15.        7+9  =  16.        8  +  9  =  17. 

9  +  9  =  18. 

Etc.  to 

Etc.  to             Etc.  to            Etc.  to 

Etc.  to 

85+9=94. 

86  +  9=95.      87+9=96.      88  +  9=97. 

89+9=98. 

These  may 

be  followed  by  tabulations  as  follows 

1—1=0. 

2—1=1.              3—1=2. 

10-1=9. 

11—1=10. 

12—1  =  11.          13—1=12. 

20—1=19. 

21—1  =  20. 

22—1=21.          23—1=22.     Etc.  to 

30—1=29. 

Etc.  to 

Etc.  to                 Etc.  to 

Etc.  to 

91—1=90. 

92—1=91.          93—1=92. 

100—1=99. 

2—2=0.             3—2=1. 

10—2=8. 

11—2=9. 

12—2=10.         13—2=11. 

20—2  =  18. 

21—2=19. 

22—2=20.         23—2=21.     Etc.  to 

30—2=28. 

Etc.  to 

Etc.  to                Etc.  to 

Etc.  to 

91—2=89. 

92—2=90.         93—2=91. 

100—2=98. 

< 

3—3=0. 

10-3  =  7. 

11-3=8. 

12—3=9.           13—3=10. 

20-3  =  17. 

21—3=18. 

22—3  =  19.         23—3=20.     Etc.  to 

30-3=27. 

Etc.  to 

Etc.  to                Etc.  to 

Etc.  to 

91—3=88. 

92—3=89.         93—3=90.                    100—3=97. 

10—4=6. 

11-4  =  7. 

12—4=8.           13—4=9. 

20—4=16. 

TWENTY  TO  GLYA'  HUNDRED.  45 


21—4  =  17. 

22—4=18. 

23-4=19. 

Etc.  to      30—  4=20. 

Etc.  to 

Etc.  to 

Etc.  to 

Etc.  to 

91—4=87. 

92—4=88. 

93—4=89. 

100—  4=90. 

10-5=5. 

11-5=0. 

12—5=7. 

13-5=8. 

20—5  =  15. 

21—  5  =  10. 

22—5  =  17. 

23—5  =  18. 

Etc.  to      30-5=25. 

Etc.  to 

Etc.  to 

Etc.  to 

Etc.  to 

91—5=80. 

92—5  =  87. 

93—5=88. 

100—5=95. 

10—0=4. 

11—0=5. 

12-0=0. 

13-0  =  7. 

20-0=14. 

21-0  =  15. 

22—0  =  10. 

23—0  =  17. 

Etc.  to      30—0=24. 

Etc.  to 

Etc.  to 

Etc.  to 

Etc.  to 

91—0=85. 

92—0=80. 

93—0=87. 

100—0=94. 

10-7  =  3. 

11-7-4. 

12—7=5. 

13—  7-=  0. 

20—  7     13. 

21—7  =  14. 

22—7-15. 

23—7-10. 

Etc.  to      30—7=23. 

Etc.  to 

Etc.  to 

Etc.  to 

Etc.  to 

91—7-84. 

92—7     85. 

93—7  -80. 

100—7=93. 

10-8-2. 

11—8=3. 

12—8     4. 

13-8-5. 

20-8  =  12. 

21-8  -13. 

22—8=14 

23-8  =  15. 

Etc.  to      30—8  =  22. 

Etc.  to 

Etc.  to 

Etc.  to 

Etc.  to 

91—8     83. 

92—8=84. 

93—8  =  85. 

100—8  =  92. 

10—9=1. 

11-9-2. 

12—9=3. 

13—9-4. 

20-9  =  11. 

21—9-12. 

22—9-13. 

23-9-14. 

Etc.  to      30—  9  =21. 

Etc   to 

Etc.  to 

Etc.  to 

Etc.  to 

91—9  =  82.         92—9  =  83.         93—9^84.  100— 9 --  =  91. 

10—10     0. 

11-10     1.        12-10-2.       13-10-3.  20-10-10. 

21 — 10  -  1 1 .     22 — 10  - 12.     23—10     1 3.    Etc.  to     30 — 10     20. 

Etc.  to  Etc.  to  Etc.  to  Etc.  to 

91—10  =  81.     92—10     82.     93—10=83.  100— 10     90. 


46  THE  SUPPL  Y. 

A  close  study  of  all  these  tabulations  should  be  made  with 
the  object  all  the  time  in  view  of  establishing  firmly  in  the 
minds  of  the  pupils  that,  for  instance,  four  and  nine  are  thirteen, 
fourteen  and  nine  are  twenty-three,  twenty-four  and  nine  are 
thirty-three,  or  in  other  words,  that  four  and  nine  added  pro- 
duce a  three  in  units'  place,  whether  they  be  simple  units  or  units 
and  tens  united.  To  this  end,  these  tabulations  should  be  on  the 
blackboard,  or  open  chart,  in  full  view  of  the  class  at  all  times; 
and  much  special  attention  should  be  devoted  to  these  com- 
binations each  day.  Certain  combinations  must  be  given  very 
much  more  attention  than  others;  for  instance,  nine  and  seven 
is  always  troublesome  and  should  receive  much  attention;  three 
and  two  is  very  simple  and  therefore  wiH  require  very  little 
attention.  But  all  must  be  dwelt  upon  until  results  are  given 
mechanically.  This  above  all  others  is  the  place  for  thorough 
work.  So  long  as  any  incorrect  results  are  announced  by  the 
pupils,  so  long  must  no  thought  of  leaving  the  subject  enter 
the  mind  of  the  teacher  or  the  pupil.  Ninety-nine  mistakes  in 
a  hundred  in  the  computations  in  arithmetic  are  the  result  of 
imperfect  knowledge  of  addition;  in  these  tabulations  is  all 
addition;  hence  the  conclusion,  that  these  concepts  must  be 
faultless,  that  _is,  drawn  from  faultless  percepts.  These  thoughts 
must  be  applied  to  column  addition,  the  real  work  of  life  for 
which  the  pupil  is  being  fitted,  if  the  public  school  system  is 
attaining  its  true  ends. 

Following  the  counting  by  tens,  by  ones,  by  twos,  by  fives, 
and  by  elevens,  should  be  formed  multiplication  tables  of  the 
same  with  one  to  ten  inclusive,  (and  from  them  division  tables 
should  be  formed)  by  the  pupils  themselves,  as  follows: 


Lxl=l 

1x2  =  2 

1x10  =  10 

1x5=5 

1x11  =  11 

2x1=2 

2x2=4 

2x10=20 

2x5  =  10 

2x11  =  22 

3x1=3 

3x2=6 

3x10=30 

3x5  =  15 

3x11=33 

4x1=4 

4x2  =  8 

4x10=40 

4x5  =  20 

4x11=44 

5x1=5 

5x2  =  10 

5x10=50 

5x5=25 

5x11=55 

6x1=6 

6x2  =  12 

6x10=60 

6x5=30 

6x11=66 

7x1  =  7 

7x2  =  14 

7x10  =  70 

7x5=35 

7x11  =  77 

8x1=8 

8x2  =  16 

8x10=80 

8x5=40 

8x11=88 

9x1=9 

9x2=18 

9x10  =  90 

9x5=45 

9x11=99 

10x1=10 

10x2=20 

10x10=100 

10x5=50 

TWENTY  TO  ONE  HUNDRED. 


47 


These  tables  may  be  read  interchangeably,  two  times  ten  are 
twenty,  two  tens  are  twenty,  two  tens  in  twenty.  The  last 
form  will  be  all  the  change  that  it  will  be  necessary  to  make 
in  passing  from  multiplication  to  division.  The  following  will 
indicate  the  co-ordinate  work  at  this  time: 
Addition: 


8 

12 

27 

21 

18 

16 

52 

<>5 

6 

2 

7 

13 

33 

19 

25 

12 

19 

8 

27 

85 

(> 

21 

25 

20 

17 

10 

9 

8 

9 

3 

4 

32 

12 

40 

31 

4<> 

12 

14 

40 

6 

Subtraction : 

17         13         16 

_9        _7        _8 

Multiplication: 
4  12 
1  1 


12 
9 


8 


13 


14 
5 


48 
_1 

21 

'> 


19 
10 


25 
12 


43 


24 


39 


4(5 

38 


6       10       12       20       18       16       14       15 
5         5         5         5         5         5         5         5 


2        3        468          230 
10      10      10      10      10      11     11      11 


5       48        791 
11     11     11      11     11     11 


Division: 

1)5        2)4 


5)15        10)40        11)33        2)16        5)35         1)9 


Applications: 

Jane  went  to  the  store  and  bought  an  orange  for  three 
cents,  some  candy  for  five  cents,  and  some  peanuts  for  four 
cents;  what  did  all  cost? 

In  one  gallon  are  four  quarts;  how  many  quarts  in  five 
gallons? 

John's  father  gave  him  ten  cents  with  which  to  buy  a  whistle 
for  five  cents,  and  to  bring  back  the  change;  how  much  did 
he  bring  back  ? 


48  THE  SUPPLY. 

I  gave  Henry  a  half-dollar  with  which  to  buy  tive  pounds 
of  sugar  at  five  cents  a  pound;  how  much  change  should  he 
bring  me  ? 

What   part   of  a   pound    of  sugar  are  eight  ounces? 

Thirty    minutes   are   what   part   of  an    hour? 

Twenty  minutes  are  what  part  of  an  hour? 

Fifteen  minutes  are  what  part  of  an  hour? 

How  much  is  a  peck  of  potatoes  worth  at  forty  cents  a 
bushel? 

A  man  riding  a  bicycle  travels  ten  miles  per  hour;  in  how 
many  hours  can  he  travel  to  Sacramento,  a  distance  of  one 
hundred  miles? 

Ten  boys  wish  to  fire  a  pack  of  fire-crackers  so  that  each 
may  fire  the  same  number;  the  pack  contains  one  hundred; 
how  many  should  each  one  fire? 

Allie  came  to  school  with  fifty  walnuts;  he  gave  each  of  his 
ten  playmates  four  walnuts;  how  many  had  he  left? 

Counting  by  fours  from  one  to  one  hundred,  beginning  with 
four,  by  threes  in  like  manner,  by  sixes,  by  eights,  by  nines, 
and  by  sevens,  should  now  receive  daily  attention  in  the  order 
given.  This  should  be  done  step  by  step.  The  fours  should 
be  thoroughly  mastered  before  the  threes  are  taken  up.  The 
counting  should  be  both  forward  and  backward.  When  taken 
in  this  order,  the  severity  of  the  work  does  not  vary  much  at 
any  two  consecutive  steps.  The  work  will  be  found  to  be 
graduated  on  the  basis  of  association  and  growth.  It  is  much 
more  difficult  to  count  by  threes  than  by  twos;  but  with  the 
graduated  progress  from  two  to  three,  here  given,  the  three 
at  its  time  will  present  no  more  difficulties  than  the  two  at 
its  time. 


ONE  HUNDRED  TO  ONE  THOUSAND. 


1OO    TO    1OOO. 


"Pythagoras    taught   that   number    is    the   first   principle   of 
existence. ' '  — Hittell. 


It  will  not  be  necessary  to  devote  much  attention  to  the 
numbers  from  one  hundred  to  one  thousand  if  those  below  one 
hundred  have  been  as  thoroughly  taught  as  they  should  be; 
for  the  mind  of  the  pupil  is  so  well  drilled  in  regard  to  the 
relative  value  of  units  and  tens  that  only  a  brief  time  will  be 
required  to  establish  the  same  relationship  between  tens  and 
hundreds.  A  single  lesson  will  often  be  sufficient  to  enable  all 
pupils  to  read  any  number  from  one  hundred  to  one  thousand. 
In  addition,  subtraction,  multiplication,  and  division,  there  is 
really  nothing  new;  simply  an  extension  of  the  identical  knowl- 
edge which  the  pupil  already  possesses.  The  work  therefore  is 
simply  expansive  in  its  nature  and  in  the  rapidity  and  accuracy 
of  its  execution. 

Much  work  of  the  following  character,  with  great  attention 
to  accuracy  and  rapidity,  must  be  given  the  pupils  every  day: 
Addition : 


198 

10 

568 

135 

532 

169 

32 

358 

139 

90 

109 

132 

9 

25 

85 

200 

26 

14 

25 

126 

9 

76 

8 

314 

75 

85 

19 

32 

7 

216 

432 

39 

110 

91 

165 

100 

Subtraction: 

865 

269 

132 

820 

736 

893 

132 

157 

129 

184 

492 

98 

Multiplication: 

231      195     263      84      36      74 
222       5      10      11 


Division: 

2)648  5)250  10)980  11)451 


50 


THE  SUPPLY. 


Tabulation   should   be  continued   as   follows: 


1x4: 
2x4: 

3X4: 
4X4  = 
5X4: 
6X4: 
7X4: 
8X4: 

9x4  = 

10X4: 


A. 
=8. 
--12. 
=16. 

=  20. 

=24. 

=28. 
-32. 
=36. 
=40. 

1x9  = 
2x9: 
3x9: 
4x9: 
5x9: 
6x9: 
7x9: 

8X9: 

9x9: 

10X9: 


1x3-3. 
2x3=6. 
3x3=9. 
4x3=12. 
5x3=15. 
6x3=18. 
7x3=21. 
8x3=24. 
9x3=27. 
10x3=30. 

:9. 

=18. 
=  27. 
=36. 
=45. 
=54. 
=63. 
=  72. 
=81. 
=90. 


1X6: 

2x6  = 
3x6: 

4X6: 

5x6  = 
6x6: 

7X6  = 

8x6: 
9x6  = 

10X6: 


:24. 
:30. 

=36. 
=42. 

=48. 
54. 
60. 

1X7: 
2X7: 
3X7: 
4X7: 

5x7: 
6x7: 

7X7: 

8X7: 

9X7: 

10X7: 


1x8=8. 

2x8=16. 

3x8=24. 

4x8=32. 
5x8=40. 
6x8=48. 
7x8=56. 
8x8=64. 
9x8=72- 
10x8=80. 

=  < . 
=14. 
=  21. 
=  28. 
=  35. 
=42. 
49. 
=56. 
=63. 
=70. 


Retain  the  readings,  two  times  three  are  six,  two  threes  are 
six,  two  threes  in  six. 

The  order  of  the  tables,  one,  two,  ten,  five,  eleven,  four, 
three,  six,  eight,  nine,  seven,  will  be  found  to  be  the  natural 
method  of  presentation,  because  the  pupil  will  learn  to  count 
by  ones  most  readily;  by  twos,  tens,  and  fives,  with  little  or 
no  difficulty;  by  elevens  nearly  as  readily;  by  fours  more  readily 
than  by  threes,  as  the  digits  in  units'  place,  four,  eight,  two, 
six,  naught,  constantly  repeat;  then  by  threes  and  by  sixes,  as 
being  next  in  order  and  naturally  associated;  by  eights,  as  being 
measurably  associated  with  fours;  by  nines,  as  being  associated 
with  threes  somewhat;  and  lastly,  by  sevens,  as  being  most 
difficult,  having  no  association  with  the  others  and  no  regularity 
of  scale  which  can  be  pointed  out  to  the  young  pupil  with 
advantage. 

Applications   should   be   continued   as   follows: 

There  are  seven  days  in  one  week;  how  many  days  in  six 
weeks  ? 

How   many   cents   in   a   dollar? 

How    many    cents   in   a   dime? 

How    many   cents    in   a   half-dollar? 


ONE  HUNDRED  TO  ONE  THOUSAND.  51 

How   many   cents   in   three   dimes? 

How   many   cents   in   a   nickel? 

How  many   cents   in    two   five-cent  pieces? 

There  are  four  quarts  in  one  gallon;  how  many  quarts  in 
six  gallons? 

There  are  three  feet  in  one  yard;  how  many  feet  in  eight 
yards  ? 

One  boy  has  eight  fingers;    how  many  fingers  have  five  boys? 

John  has  two  cents  and  James  has  one-half  as  many;  how 
many  has  James? 

What   is   one-half  of  four? 

How   many   cents   in   one   dime? 

How   many   cents   in   one  half-dime? 

How   many    pints   in    one  quart? 

How    many   pints   in    one-half  of  a    quart? 

I  gave  Julia  six  cents  and  Mary  one-half  as  many;  how  many 
did  I  give  Mary? 

Hattie  brought  me  three-  pinks  and  Lily  brought  me  two  less 
than  Hattie;  how  many  did  both  bring  me? 


CO-ORDINATE  EXERCISES. 


4-  Of  2=  = 

2-s-2=  3-5-3  =  5-5  = 

.Vof4=  |  of  6=  ioflO  = 

4-5-2=  6-5-3=  10-«-5= 

4- of  6=  iof9  =  ±of!5  = 

6-i-2=  9-t-3  =  15-5-5  = 

£of8=  fofl2=  \  of  20= 

8-*-2  =  12--3=                 Etc.  to  20-i-5= 

4- of  10=  £of!5  =  -I  of  23  = 

10—2=  15-s-3=  25+5= 

Etc.  to  Etc.  to  Etc.  to 

4-of20  =  i  of  30=  }  of  50= 

20-s-2=  30-i-3=  50—5  = 

128  703         862  83 

93  9        -195  xll        10)840 

75  83 

46  12         708  95 

28  109         —95  X5         5)625 


THE  SUPPLY 


31 

20 

105 


500 
-180 


87 
X2 


11)891 


Preparatory  to  ready  and  thorough  work  in  "  Long  Division," 
in  Factoring,  and  in  Greatest  Common  Divisor  and  Least  Com- 
mon Multiple,  the  following  chart  should  be  made  by  the  class 
under  the  direction  of  the  teacher: 


1 

o 

3 

4 

5 

8 

7 

8 

9 

10 

11 

12  . 

13 

14 

15 

16 

17 

18 

9 

4 

8 

8 

10 

12 

14 

16 

18 

20 

•2-2 

2-1 

2U 

28 

30 

32 

31 

36 

3 

6 

9 

12 

15 

18 

21 

24 

27 

30 

33 

36 

39 

42 

45 

48 

51 

54 

4 

8 

12 

16 

20 

24 

28 

32 

36 

40 

44 

48 

52 

.")(i 

60 

64 

68 

72 

6 

10 

15 

20 

25 

30 

35 

40 

45 

.50 

55 

6!) 

65 

70 

75 

80 

85 

90 

(i 

12 

18 

24 

30 

36 

42 

48 

54 

60 

66' 

72 

78 

84 

90 

96 

102 

108 

i 

14 

21 

28 

35 

42 

49 

56 

63 

70 

77 

84 

91 

98 

105 

112 

119 

I2(i 

8 

16 

24 

32 

40 

48 

56 

64 

72 

80 

88 

06 

104 

112 

12'1 

128 

13(i 

1-14 

9 

18 

27 

36 

-15 

54 

63 

72 

81 

90 

99 

los 

117 

126 

1.35 

144 

153 

162 

10 

20 

30 

40 

50 

60 

70 

80 

90 

100 

110 

12.  > 

130 

140 

150 

160 

170 

180 

11 

22 

33 

44 

55 

66 

77 

88 

99 

110 

121 

132 

148 

187 

12 

24 

86 

48 

60 

72 

84 

96 

108 

12) 

132 

144 

13 

26 

89 

52 

65 

78 

91 

104 

117 

130 

143 

169 

221 

14 

28 

42 

58 

70 

84 

9.S 

112 

12i 

1-10 

196 

15 

80 

45 

BO 

75 

80 

105 

120 

135 

1.50 

225 

16 

32 

4* 

64 

SO 

M 

112 

12-! 

141 

160 

256 

17 

84 

51 

68 

65 

102 

11!) 

136 

153 

170 

1S7 

221 

289 

18 

88 

54 

72 

!»() 

10S 

126 

144 

162 

180 

824 

10 

38 

57 

76 

95 

114 

133 

152 

171 

191) 

219 

247 

323 

ONE  THOUSAND  AND  Aim  YE. 


1OOO   AND  ABOVE. 


"It    is  comparatively  rare  to   find  a  candidate  who   can    add 
correctly  a  long  column  of  figures." — Civil  Service  Examiners. 


In  order  that  the  true  conception  of  the  method  of  reading 
and  writing  large  numbers  may  be  obtained  by  the  pupils,  it 
will  be  found  simplest  to  make  inclosures  of  some  kind  in  which 
the  numbers  may  be  written  by  periods  of  three  figures  each, 
each  inclosure,  or  period,  being  read  as  if  standing  alone,  the 
pupils  attaching  to  the  reading  of  the  number  in  the  period  the 
name  of  the  enclosure,  or  period. 

The  following  will  illustrate  the  method  and  convince  the 
reader  of  its  simplicity; 


865 
1000 
85  5H5 
732  840 
4  625  835 


Much  practice  should  be  given  to  the  reading  of  the  numbers 
in  the  several  boxes,  or  periods,  separately  at  first,  having 
definite  names  for  the  boxes  in  accordance  with  their  location. 
The  right  hand  box  may  be  left  nameless,  as  the  numbers 
therein  are  always  read  without  the  box,  or  period,  name. 
The  numbers  in  the  second  box  (thousands)  should  be  read 
without  any  reference  to  the  other  boxes:  that  is,  the  85  of  the 


54  THE  A  UPPL  Y. 

number  S5,5(>5  should  be  read  as  eighty-five,  and  the  word 
"thousands"  should  attach  simply  as  the  word  "horses"  would 
attach  to  the  number.  In  this  way  the  reading  of  numbers  is 
very  simple,  as  no  numeration  is  ever  indulged  in  to  retard 
rapidity  of  execution.  After  this  process  of  reading  is  carried 
on  until  numbers  are  read  without  the  least  hesitation,  the  box 
names  may  be  removed  and  the  readings  continued  until  the 
names  are  fixed  by  location,  when  the  boxes  themselves  may 
be  omitted  and  the  comma  used  in  their  stead.  The  comma 
should  never  be 'omitted  afterward  until  both  the  reading  and 
the  writing  of  all  numbers  is  as  rapid  and  accurate  as  possible. 
After  the  reading  of  numbers  in  the  several  steps  given,  the 
pupil  should  be  given  much  practice  in  the  writing  of  numbers 
in  the  several  steps  successively.  The  method  indicated  will 
obviate  the  difficulty  and  doubt  encountered  by  all  in  knowing 
when  to  insert  ciphers  and  how  many,  and  will  certainly  save 
much  labor  and  induce  accuracy  and  rapidity. 

Work  now  should  broaden  out  to  its  fullest  in  both  addition 
and  subtraction.  The  subject  ot  addition  should  at  all  times 
be  given  the  most  careful  attention;  pupils  should  be  taught 
to  add  upward  and  downward  with  equal  and  great  facility, 
naming  results  only  as  they  proceed.  The  time  devoted  to  ad- 
dition as  compared  to  that  devoted  to  subtraction  must  be, 
thoughtfully  speaking,  as  ten  to  one;  and  nearly  the  same 
ratio  will  be  found  to  exist  between  the  time  required  for  ad- 
dition and  that  required  for  either  multiplication  or  division. 
Nearly  all  errors  in  school  and  among  business  men  are  directly 
or  indirectly  the  result  of  imperfect  work  in  addition.  Pupils 
should  add  long  columns,  with  a  view  to  accuracy  first  and 
rapidity  secondly,  every  day  of  their  school  lives  until  they  are 
twelve  years  old,  and  even  then  the  exercise  must  not  be  allowed 
to  fall  into  any  considerable  degree  of  disuse. 

Multiplication  by  numbers  containing  two  or  more  digits 
should  also  be  practiced  at  this  stage  of  advancement,  and 
clear  ideas  in  regard  to  the  beginning  of  a  partial  product  one 
place  farther  to  the  left  than  its  immediate  predecessor  should 
be  inculcated. 

Short  division  of  all  numbers  should  be  given  much  attention. 

The  following  indicates  the  character  of  the  co-ordinate  work 
at  this  time: 


ONE  THOUSAND  AND  ABOVE.  55 

Addition: 

8,963       89,645       986,746       456,834       84 

7,054       38,395       300,846       904,386       36 

765       26,283       258,984       784,666       92 

392       13,725       98, 642       292,564       78 

843      185.939        89,206       964,238       43 

6. 483       205.684      1.864.684       792. 189       56 

29 
46 
92 
_73 

Subtraction : 

189,425  40,389  $36,472  4,007,392 

74.932  19,426  293. 783  1.006.829 

8,965,432  7,064,584  34,856,329  867,345,732 

2.893,876  2.092.864  19.857.184  193.845.783 

Multiplication: 

64.530,243  894,320  5,004,234  74,385 

25  360  125  170 


986.432,894  846.785  2,345,860  93,656,487 
432            79_  280             564 

Division: 

6)482. 356  8)8.375.493  9)84.293.008  7)483.754 

Constant  review  of  tabulations  regularly  and  irregularly,  oral 
and  written:    and  further  extensions  of  tabulations  as  follows: 

1x12  =  12.  1x13  =  13.  1x14  =  14.  1x15  =  15. 

2x12  =  24.  2x13  =  26.  2x14=28.  2x15=30. 

3x12=36.  3x13=39.  3x14=42.  3x15=45. 

4  x  12  =  48.  4  x  13  =  52.  4  x  14  =  56.  4  x  15=60. 

5x12=60.  5x13=65.  5x14  =  70.  5x15=75. 

6x12  =  72.  6x13  =  78.  6x14=84.  6x15  =  90. 

7x12-84.  7x13  =  91.  7x14  =  98.  .7x15  =  105. 

8x12=96.  8x13  =  104.  8x14  =  112.  8x15  =  120. 


THE  Sr 


9x12  =  1  OS.  9  x  1.3  =  117.  9x14  =  1  26. 

10x12  =  120.         10x13  =  130.         10x14  =  140, 

1x16  =  16, 
2x16=32. 
3x16=48. 
4x16-64. 
5x16=80. 
6x16=96. 
7x16=112. 
8x16  =  128. 
9x16  =  144. 
10x16=160. 


9x15  =  185. 
10x15  =  150. 


1x17=17. 

1x18  =  18. 

1x19  =  19. 

1x20=20. 

2x17=34. 

2x18  =  36. 

2x19=38. 

2x20=40. 

3x17=51, 

3x18=54. 

3x19  =  57. 

3x20  =  60. 

4x17  =  68. 

4x18=72. 

4x19  =  76. 

4x20=80. 

5x17=85. 

5x18=90. 

5x19=95. 

5x20=100. 

6x17  =  102. 

6x18  =  108. 

6x19=114. 

6x20=120. 

7x17  =  119. 

7x18=126. 

7x19  =  133. 

7x20=140. 

8x17=136. 

8x18=144. 

8x19  =  152. 

8x20=160. 

9x17  =  153. 

9x18  =  162. 

9x19=171. 

9x20  =  180. 

10x17  =  170. 

10x18=180. 

10x19  =  190. 

10x20  =  200. 

The  preceding  tables  should  not  be  directly  committed  to 
memory  but  should  be  given  to  the  pupils  with  the  last  columns 
blank,  and  the  pupils  should  be  required  to  fill  in  the  products 
on  their  slates  or  blank  books  or  upon  the  blackboard.  In 
this  way  much  greater  power  will  be  acquired  for  "long  divi- 
sion" than  could  be  acquired  without  these  exercises,  and  ac- 
curacy and  rapidity,  the  watch-words  of  arithmetic,  will  be  given 
a  wonderful  impulse. 

The  applications  should  now  be  more  comprehensive,  and 
simple  explanations  should  be  developed.  These  applications 
should  include  more  and  more  the  tables  of  compound  num- 
bers that  are  of  frequent  use  in  the  every  day  affairs  of  life. 
In  developing  explanations,  begin  with  simple  applications  like 
the  following: 

What  will'  four  apples  cost  at  one  cent  each? 

Explanation:  Since  one  apple  costs  one  cent,  four  apples 
will  cost  four  times  one  cent,  or  four  cents. 


ONE  THOUSAND  AND  ABOVE.  f>7 

Keep  the  explanations  short  yet  complete. 

Another:  How  many  oranges  can  I  buy  with  ten  cents,  at 
two  cents  a  piece? 

Explanation:  Since  one  orange  costs  two  cents,  for  ten  cents 
I  can  buy  five  oranges. 

This  explanation  is  short  and  pointed,  but  is  objected  to  some- 
times as  failing  to  indicate  the  process.  Later  on  the  clause 
indicating  the  process  may  be  introduced.  The  explanation 
will  then  be  as  follows:  Since  one  orange  costs  two  cents,  I 
can  buy  as  many  times  one  orange  for  ten  cents  as  two  cents 
are  contained  times  in  ten  cents,  or  five  times  one  orange,  or 
five  oranges. 

The  following  explanation  is  also  a  good  one,  but  is  open  to 
the  objection  that  it  is  not  consonant  with  the  general  process 
employed  by  the  business  world:  Since  two  cents  will  buy  one 
orange,  one  cent  will  buy  one-half  an  orange  and  ten  cents  will 
buy  ten  times  one-half  an  orange,  or  five  oranges. 

This  explanation  is  also  open  to  the  objection  that  it  involves 
the  use  of  fractions  which  are  as  yet  but  imperfectly  developed. 
A  concise  and  precise  explanation  is  of  double  utility;  it  assists 
the  understanding  and  begets  confidence,  and  at  the  same  time 
is  incidentally  a  valuable  language-builder. 


THE  SUPPLY. 


THE    PRIME    NUMBER. 


The  concept  of  a  prime  number  is  a  necessary  antecedent  to 
a  clear  conception  of  the  subjects  of  Greatest  Common  Divisor 
and  Least  Common  Multiple.  The  method  by  which  this  concept 
is  implanted  in  the  mind  of  the  pupil  is  both  simple  and  un- 
failing: 

T.     (To  class. )     By  what  may  one  be  divided  ? 
C.     By   one. 

T.     By   what   may   two   be   divided? 

C.     By   two   and    by   one.  1  4 

T.     By  what  may   three  be   divided?  2  <> 

C.     By   three   and   by   one.  H  H 

T.      By   what   may   four   be   divided?  5  ?> 

C.     By   four,    by  two,    and   by   one.  7  10 

T.  Do  you  notice  any  difference  between  the  four  and  the 
three  ? 

C.  Yes;  four  may  be  divided  by  some  other  number  than 
itself  and  one. 

T.     Then  we  shall  place  it  in  another  column. 

T.     By  what  numbers  may  five  be  divided?          11  \"2 

C.     By   five   and   by    one,  13  14 

T.     Then  in  which  column  shall  I  write  it?  17  15 

C.     In  the   first   column.  1<> 

T.      By   what   may  six    be  divided? 

C.     By   six,    by   three,    by   two,    and   by   one. 

T.     Does   six   belong   in   the   first   column  then? 

C.     No. 

T.     That  is  right;    and  as  we  will  write  all  numbers  in    the 


THE  PRIME  NUMBER.  oi> 

second  column  that  do  not  belong  in  the  first  column,  I  shall 
write  this  one  in  the  second  column. 

T.      By   what   may   seven    be   divided? 

C.     By   seven   and   by   one. 

T.      In   which    column    shall    I    write    it? 

C.     In  the   first. 

T.      By   what    may    eight   be   divided? 

C.     By   eight,    by   four,    by   two,    and    by   one. 

T.     James,    you    may   tell    me    in  which  column   to  write  it. 

J.       In    the   first. 

T.      Lucy,    did  James   answer   correctly? 

L.     No;    it   should   be   written    in    the   second   column. 

T.     Can    you    tell    me   why? 

L.  Because  it  may  be  divided  by  other  numbers  than  eight 
and  one. 

T.     Very   good. 

T.     Mary,    by   what   may   nine   be   divided? 

M.     By   nine,    by   three,   and    by    one. 

T.     To   which    column    does   it   belong? 

M.    To  the  second. 

T.     To   which   column   then   do   you    think   ten   belongs? 

M.  I  think  ten  belongs  to  the  second  column,  because  it 
may  be  divided  by  ten,  by  five,  by  two,  and  by  one. 

T.     Now  who  can  tell   to   which   column   eleven   belongs? 

Robert.  (Holding  up  his  hand.)  It  belongs  to  the  first, 
because  it  may  be  divided  by  eleven  and  one  only. 

T.     That   is  right. 

T.  The  numbers  in  the  first  column  are  called  PRIME 
NUMBERS. 

T.  Now  think  a  moment,  and  then  I  want  all  who  think 
they  can  tell  me  what  a  prime  number  is  to  raise  their  right 
hands. 

T.  (After  waiting  a  short  time.)  How  many  think  they 
can  tell  me  what  a  prime  number  is? 

(Many  hands  are  raised  and  after  many  answers  more  or 
less  perfect.) 

Robert.  A  prime  number  is  a  number  that  may  be  divided, 
without  a  remainder,  by  itself  and  one  only. 

T.  Jane,  you  may  tell  me  whether  twelve  is  a  prime  number 
or  not? 


<>0  THE  SUPPLY. 

(.  Twelve  is  not  a  prime  number,  because  it  may  be  divided 
by  some  other  number  than  itself  and  one,  by  six  for  example. 

T.  Helen,  you  may  tell  me  whether  thirteen  is  a  prime 
number  or  not? 

H.  Thirteen  is  a  prime  number,  because  it  may  be  divided 
by  itself  and  one  only. 

T.  Thomas,  you  may  write  fourteen,  fifteen,  sixteen,  and 
seventeen  in  these  columns,  being  careful  to  place  each  num- 
ber where  it  belongs. 

In  the  same  manner  all  numbers  from  one  to  one  hundred 
should  be  segregated  into  the  two  classes,  prime  numbers 
and  those  that  are  not  prime  numbers.  The  prime  numbers 
then  should  be  committed  to  memory  as  soon  as  is  convenient, 
so  that  errors  in  greatest  common  divisor  and  least  common 
multiple  may  be  avoided. 


O.  C.  D.  AND  L.  C.  M.  in 


GREATEST  COMMON  DIVISOR  AND 
LEAST  COMMON  MULTIPLE. 


' '  I  assure  you  there  is  no  such  whetstone  to  sharpen  a  good 
wit  and  encourage  a  will  to  learning  as  is  praise." — Ascham. 


The  object  of  developing  the  idea  of  a  prime  number  was 
to  prepare  the  student  properly  to  comprehend  the  subjects 
of  greatest  common  -divisor  and  least  common  multiple  These 
subjects  should  be  approached  only  by  means  of  factoring; 
that  is,  by  considering  numbers  with  reference  to  the  prime 
numbers  that  are  contained  in  those  numbers  an  exact  num- 
ber of  times.  For  example,  4-2x2. 

T.     What   is    a   divisor   of  four? 

C.     Two    is   a   divisor   of  four. 

T.     What    is   the   greatest   divisor   of  four? 

C.     Two    twos,    or   four,    is    the   greatest   divisor  of  four. 

Then  take  two  numbers  and  resolving  them  into  their  prime 
factors: 

<>-2x.S. 
8  =  2x2x2. 

T.  What   is   a   divisor  of  six? 

C.  Two    is   a  divisor  of  six. 

T.  What    is   another   divisor   of  six? 

C.  Three   is   another   divisor   of  six. 

T.  Is   there   any    other   divisor   of  six? 

C.  Yes;    two   times   three,    or  six,    is   another   divisor. 


<;:>  THE  SUPPLY. 

T.     What   is   a   divisor   of  eight? 

C.     Two,    two  times   two,    or   two   times   two   times   two. 
T.     Which    of  these   is   also   a   divisor   of  six? 
C.     Two   is   also   a   divisor   of  six. 

T.      Yes;    two  is  also  a  divisor  of  six.     Two  then  is  a  common 
divisor   of    six    and    eight. 

Then    take   three  or   more   numbers   that  contain  more  than 
one   common   divisor: 

6-2  x  8. 

12=2x2x3. 

18-2x3x3. 

T.     What   are   the   divisors  of  six  ? 

C.     Two,  three,  and  two  times  -three  are  the  divisors  of  six. 

T.     What   are   the   divisors   of  twelve? 

C.  Two,  three,  two  times  two,  two  times  three,  and  two 
times  two  times  three  are  the  divisors  of  twelve. 

T.     What   are   the   divisors   of  eighteen? 

C.  Two,  three,  two  times  three,  three  times  three,  and  two 
times  three  times  three  are  the  divisors  of  eighteen. 

T.  Now  look  carefully  and  see  what  will  divide  each  one 
of  the  three  numbers. 

C.     Two   will   divide  each   one   of  the    numbers. 

T.     What   else  will   divide   each   one   of  the    numbers? 

C.     Three   will   also   divide   each   one   of  the   numbers. 

T.  Is  there  any  other  number  beside  two  and  three  that 
will  divide  each  one  of  the  numbers,  six,  twelve,  and  eighteen? 

C.     Yes;    two   times   three   will   also   divide   them. 

T.  Then,  what  are  the  common  divisors  of  six,  twelve, 
eighteen  ? 

C.  Two,  three,  and  two  times  three,  or  six,  are  the  common 
divisors  of  six,  twelve,  and  eighteen. 

T.  Then,  what  is  the  greatest  common  divisor  of  six,  twelve, 
and  eighteen? 

C.  Two  times  three,  or  six,  is  the  greatest  common  divisor 
of  six,  twelve,  and  eighteen. 

T.  Now,  looking  at  the  prime  factors  of  six,  twelve  and 
eighteen,  can  you  tell  me  what  the  greatest  common  divisor 
of  the  numbers  contains? 

C.     The  greatest  common  divisor  of  six,  twelve,  and  eighteen, 


G.  C.  D.  AND  L.  C.  M.  63 

contains  all  the  the  prime  factors  that  are  found  in  each  of 
them. 

T.  I  wish  to  see  now  whether  you  can  find  the  greatest 
common  divisor  of  fifteen,  twenty-five,  thirty-five,  and  forty-five. 

C.  The  prime  factors  of  fifteen  are  three  and  five;  the  prime 
factors  of  twenty-five  are  five  and  five;  the  prime  factors  of 
thirty-five  are  five  and  seven;  and  the  prime  factors  ~or  forty- 
five  are  three,  five,  and  five.  Five  is  the  only  factor  that  is 
contained  in  each  of  the  numbers,  fifteen,  twenty-five,  thirty- 
five,  and  forty-five;  hence  five  is  the  greatest  common  divisor 
of  fifteen,  twenty-five,  thirty-five,  and  forty-five. 

Blackboard    form :  15  =  3  x  5. 

25  =5x5.  • 
35^5x7. 
45-3x3x5. 
Therefore   5  =  G.    C.    D. 

The  visible  solution  should  be  as  brief  as  possible,  consistent 
\vith  a  clear  understanding  of  the  process.  The  solution  should 
be  accompanied  or  supplemented  by  an  oral  explanation  that 
shall  convince  both  the  teacher  and  the  pupils  that  the  pupil 
clearly  understands  his  work.  The  idea  of  the  greatest  com- 
mon divisor  being  developed  and  clearly  and  firmly  fixed  in 
the  minds  of  the  pupils,  the  subject  of  the  least  common  mul- 
tiple should  be  approached  by  an  analogous  process: 

T.     What   number   contains   two  an  exact  number  of  times? 

C.      Four   contains   two   twice. 

T.  Is  there  any  other  number  that  contains  two  an  exact 
number  of  times? 

C.     Yes;   six  and   eight,  as   well   as  a   great   many   others. 

T.     What  number  contains  three  an  exact  number  of  times  ? 

C.     Six,    nine,    twelve,    and   many   other   numbers. 

T.  As  six  contains  three  an  exact  number  of  times,  it  is 
called  a  multiple  of  three. 

T.     Now   what   is   a   multiple   of  four? 

C.  Eight  is  a  multiple  of  four;  twelve  is  a  multiple  of  four: 
many  other  numbers  are  multiples  of  four. 

T.  What  number  is  a  multiple  of  two  and  also  a  multiple 
of  three  ? 

C.     Six  is  a  multiple   of  two   and  also  a    multiple   of  three. 


(>4  THE  SUPPLY. 

T.     What   did   we   call   a   number   that    is   a.  divisor  of  two 
or   more   numbers? 

C.     We   called   a   number   that   is  a  divisor  of  two  or  more 
numbers,  a   common   divisor. 

T.     Then    what   shall   we   call   a    number   that   is   a  multiple 
of  two    or   more   numbers? 

C.     A    number  that   is   a    multiple  of  two  or  more  numbers 
is   a   common  multiple   of  those   numbers. 

T.     Then   six    is   a   common   multiple   of  two   and   three. 

T.     Now,    can    you    think  of  any  other  common  multiple  of 
two  and   three  ? 

C.     Twelve    is   another  common    multiple  of  two  and  three; 
eighteen    is   another   common    multiple   of  two   and   three. 

T.     What   is   the  least   common  multiple  of  two  and  three? 

C.     Six  is  the  least  common  multiple  of  two  and  three. 

Take  numbers  and  factor  them  into  their  prime  factors: 
6=2x3. 
8=2x2x2. 
9=3x3. 

T.     What   factors   does   six    contain? 

C.     Six    contains   two   and   three. 

T.     Then   a   number    that   contains    six    must   contain  what? 

C.     A   number   that   contains   six    must    contain    the   factors 
two   and   three. 

T.     A   number   that   contains,  eight   must   contain   what? 

C.     A   number   that   contains   eight    must   contain   two  as    a 
factor   three   times. 

T.     Then   a   number  that   contains  both  six  and  eight  must 
contain   what  ? 

C.     A   number   that   contains   both  six  and  eight  must  con- 
tain,   as   factors,   three   twos   and   a   three. 

T.     What  number  contains  three  twos  and  a  three  as  factors? 

C.     Twenty-four  contains  three  twos  and  a   three  as  factors. 

T.     Then   what   is   a   common  multiple  of  six  and  eight? 

C.     Twenty-four   is  a  common  multiple  of  six   and   eight. 

T.     Is   there   any   other   number   less   than   twenty-four  that 
contains   both   six   and   eight? 

C.     No. 

T.     Then   what    is    the   least   common   multiple   of  six    and 
eight  ? 


G.  C.  D.  AND  L.  C.  V.  tio 

C.  Twenty-four  is  the  least  common  multiple  of  six  and 
eight. 

T.  Now  look  carefully  at  the  three  numbers,  six,  eight,  and 
nine,  and  tell  me  what  factors  the  least  common  multiple  of 
these  three  numbers  must  contain? 

C.  The  least  common  multiple  of  six,  eight,  and  nine  must 
contain  three  twos  and  two  threes  as  factors;  since  to  contain 
eight  it  must  contain  three  twos,  and  to  contain  nine  it  must 
contain  two  threes;  to  contain  the  six  no  additional  factors 
are  necessary  as  already  all  of  its  factors  have  been  selected. 

T.  Then,  the  least  common  multiple  of  six,  eight,  and  nine 
is  what  ? 

C.  The  least  common  multiple  of  six,  eight,  and  nine  is 
seventy-two. 

T.  Now  you  may  find  the  least  common  multiple  of  ten, 
fifteen,  twenty,  and  twenty-five. 

C.     10  =  2x5. 
15=3x5. 
20=2x2x5. 
25  =  5x5. 

Therefore    2x2x3x5xo=300=L.    C.    M. 

The  least  common  multiple  must  contain  as  factors  two  twos, 
because  it  must  contain  all  the  prime  factors  of  each  number 
and  there  are  two  twos  in  twenty;  it  must  contain  a  three 
in  order  that  it  may  contain  fifteen;  and  it  must  contain  two 
fives  in  order  that  it  may  contain  twenty-five.  It  need  not 
contain  any  other  factors,  for  it  now  contains  all  the  prime 
factors  of  each  of  the  numbers;  hence  the  least  common  mul- 
tiple of  ten,  fifteen,  twenty,  and  twenty-five  contains  as  factors 
two  twos,  one  three,  and  two  fives,  and  is  therefore  three 
hundred. 


THE  SUPPL  Y. 


FRACTIONS, 


•  "Questions  must  be  given  to  suit  all  capacities."- — Currie. 


While,  in  considering  whole  numbers,  we  may  gain  for  the 
pupil  some  '  idea  of  fractions,  the  true  idea  of  fractions  is  never 
obtained  until  the  unit  is  separated  into  its  parts.  Usually 
children  before  coming  to  school  have  an  indefinite,  and 
perhaps  •  in  a  few  exceptional  cases  a  definite,  idea  of  certain 
fractional  parts,  as  halves  and  quarters;  it  is  very  rare  that 
any  other  fractional  parts  are  familiar  to  any  of  them.  It 
will  generally  be  found,  moreover,  that  their  idea  of  a  half 
is  only  approximate,  that  is,  that  they  consider  that  an  apple 
divided  into  two  parts  in  the  ordinary  way  is  divided  into 
halves;  hence  the  use  of  the  apple  or  of  any  other  irregularly 
formed  object  must  be  supplemented  by  something  more  exact 
in  its  dimensions.  The  line  upon  the  blackboard,  as  it  may 
be  measured  by  the  pupil  himself,  is  probably  the  best  object 
with  which  to  correct  the  pupil's  crude  definition  of  a  half, 
(to-wit:  A  half  is  one  of  the  two  parts  of  any  thing.)  and 
to  induce  him  to  insert  the  word  "equal"  between  "two"  and 
"parts"  and  to  make  other  corrections  if  necessary,  giving 
ultimately  the  true  definition,  A  half  is  one  of  the  two  equal 
parts  of  any  thing. 

The  following  outline  of  a  first  lesson  in  fractional  parts  will 
indicate  the  character  of  the  work  deemed  proper,  the  quantity 
being  left  to  the  good  judgment  of  the  faithful  an  inventive 
teacher: 

T.  (Holding  up  an  apple  and  cutting  it  into  two,  as  nearly 
as  possible,  equal  parts.)  Into  how  many  parts  have  I  cut  the 
apple? 


FRACTIONS.  67 

C.     You    have   cut   it   into   two   parts. 

T.     What   is   each   part? 

C.      Each    part   is  one-half. 

T.  .   That   is   right;    th'en   what    is   one-half? 

C.     It   is  one   of  the   two   parts   of  any   thing. 

T.  Now  look  closely  and  I  will  take  another  apple  and 
cut  it.  (Cutting  the  apple  into  two  parts  quite  unequal  in 
size.)  Is  each  one  of  these  parts  a  half? 

C.      (Answers   at   first   are   divided.) 

T.  I  will  take  another  apple  and  cut  it.  (Cutting  it  into 
two  parts  very  unequal  in  size.)  Is  each  one  of  these  parts 
a  half? 

C.     (Seeing   now   the  great   inequality.)     No. 

T.      Why    not? 

C.     The   parts    are    not  equal. 

T.  But  you  said,  One-half  is  one  of  the  two  parts  of  any 
thing.  Do  you  wish  to  correct  your  definition? 

C.     Yes. 

T.  Very  well,  you  may  do  so  if  you  will  be  very  careful. 
Ready. 

C.     A    half  is   one   of  the   two  equal    parts  of  any  thing. 

T.     That   will   do   very   well. 

T.     How    many   halves    in  any  thing? 

C.     There   are   two    halves   in    any   thing. 

T.  I  will  write  one-half  on  the  blackboard.  Q-.)  The  two 
under  the  line  is  the  number  of  parts  into  which  I  divided 
the  apple,  and  the  one  is  the  one  part. 

T.     How    many   halves   in    one? 

C.     There   are  two   halves  in   one. 

T.     How    many   are   two   times    one-half? 

C.     Two   times   one-half  are   one. 

T.     How   much   are   one-half  and   one-half? 

C.     One-half  and  one-half  are  one. 

T.     If  I    take   away   one-half  from  one,   how   much    is   left? 

C.     One-half  is   left. 

T.     How   shall    I    write  two-halves? 

C.     f. 

T.  If  I  take  two-halves  of  one  apple  and  one-half  of  another, 
how  many  halves  will  I  take? 

C.     You   will   take   three-halves. 


OS  THE  SUPPLY. 

T.      How   shall    I    write   three-halves   on    the   blackboard? 
p      3 

L-.  TjT. 

T.     How    much    are   two-halves   and    one-half? 

C.     Two-halves   and   one-half  are  tfiree-halves. 

T.      How    much   are  three   times   one-half? 

C.     Three   times   one-half  are   three-halves. 

T.      How  many  times  is  one-half  contained  in  three-halves? 

C.     One-half  is  contained   in  three-halves  three  times. 

T.     How   much    is   two-halves? 

C.     Two-halves    is   one. 

T.     How    much    is   one   plus   one-half? 

C.     One   plus   one-half  equals   one-and-one-half. 

T.     This   is   the   way    one-and-one-half  is   written.     (14-. ) 

T.      How    much    then    are   two-halves  and  one-half? 

C.     Two-halves   and  one-half  are  one-and-one-half. 

T.      How   much   then    is  three-halves? 

C.     Three-halves   equals    one-and-one-half. 

The  subject  of  halves  may  be  continued  in  this  manner  until 
the  teacher  feels  satisfied  that  the  subject  is  understood. 

The  subject  of  thirds  should  be  presented  upon  the  black- 
board with  the  line,  in  substantially  the  same  manner  that  the 
subject  of  halves  was  presented  with  the  apple. 

T.      Into    how    many   parts    is   this   line   divided? 


C.      It   is   divided   into   three   parts. 

T.  Measure  the  parts  and  tell  me  into  what  kind  of  parts 
it  is  divided. 

C.     It   is   divided    into   three   eq^^al  parts. 

T.     What   shall   we   call   one   of  the   parts? 

(The   teacher   may   be  obliged  to  give   the  name  himself.) 

T.    or   C.     One-third. 

T.      How  shall    I    write   it? 

C.     i. 

T.     What    is  one-third? 

C.     One-third   is  one  of  the  three  equal  parts  of  any  thing. 

The  subject  of  thirds  may  be  continued  as  the  subject  of 
halves  was  continued.  No  definite  comparisons  between  the 
halves  and  the  thirds  should  be  made  until  the  subject  of  sixths 
has  been  considered. 


FRACTIONS*  68 

The  subject  of  fourths  should  be  approached  in  the  same 
manner  as  the  subject  of  thirds,  and  requires  the  same  character 
of  questions  and  work,  with  the  following  in  addition: 

T.      Into    how  many   parts  have    I    divided   this   line? 


C.     Into  two   parts, 
T.     What   is  each   part? 
C.      Each    part   is  one- half. 

T.     (Using   the  same  line.)      Into   how  many   parts   have    I 
divided   it   now? 


C.     Into   four  parts. 

T.     What   is   each   part? 

C.      Each    part   is  one-fourth. 

T.     How   many   one-fourths   in    one-half? 

C.     There   are   two   one-fourths   in  one-half. 

T.     What   is   one-half  of  one-half? 

C.     One-half  of  one-half  is   one-fourth. 

T.     One-half  equals   what? 

C.     One-half  equals  two-fourths. 

T.     How  much  are  one-half  and  one-fourth? 

C.     One-half  and   one-fourth  are   three-fourths. 

At  this  point  practical  applications  in  .fractional  forms  should 
be  introduced  that  involve  the  ideas  of  halves  and  fourths  either 
separately  or  conjointly. 

The  following  will  indicate  something  of  the  nature  of  the 
applications  that  may  be  given  advantageously: 

(Short  oral  explanations  should  accompany  all  applications.) 

What  is  the  sum  of  one-half  of  four  cents  and  one-fourth  of 
four  cents  ? 

What   is   three-fourths   of  four   cents? 

If  a  pound  of  coffee  is  worth  sixteen  cents,  what  is  one-fourth 
of  a  pound  worth  ? 

If  a  gallon  of  milk  cost  twenty  cents,  what  will  a  quart  cost? 

How    many    hours   in   a   day? 

How   many    hours  in    one-half  of  a   day? 

How   many   hours   in   one-fourth   of  a   day? 


70  THE  SUPPLY. 

How   many   hours   in   three-fourths   of  a   day? 

How    many   cents   in   one   dollar? 

How    many   cents   in   one- half  of  a   dollar? 

How   many   cents    in    one-fourth    of  a   dollar? 

How   many   cents   in   three-fourths   of  a   dollar? 

How   many   ounces   in   one   pound   of  sugar? 

How   many   ounces    in    one-half  of  a   pound  of  sugar? 

How    many   ounces   in    one-fourth    of  a   pound  of  sugar? 

How   many   ounces   in  three-fourths  of  a  pound  of  sugar  ? 

There  are  thirty-one  and  one-half  gallons  in  a  barrel;  how 
many  gallons  in  two  barrels? 

There  are  five  and  one-half  yards  in  one  rod;  how  many 
yards  are  there  in  two  rods? 

There  are  four  quarts  in  a  gallon;  one  quart  is  what  part  ot 
a  gallon  ? 

Two  quarts  are  what  part  of  a  gallon  ? 

Three  quarts  are  what  part  of  a  gallon  ? 

The  subject  of  fifths  is  entirely  analogous  to  that  of  thirds  in 
the  manner  of  its  presentation. 

The  subject  of  sixths  is  quite  like  that  of  fourths,  except  that 
it  requires  the  establishment  of  the  equivalence  in  sixths  of  both 
thirds  and  halves,  as  well  as  more  extensive  applications  in- 
volving the  three  fractional  units. 

The  following  will  give  a  general  idea  of  the  manner  of  de- 
veloping this  additional  conception: 

T.     Into  how  many  parts  is  this  line  divided? 


C.  Into  two  parts. 

T.  What  is  each  part? 

C.  One-half. 

T.  Into  how  many  parts  is  the  line  divided    by   the   points 

below  the  line? 

I 


C.     Into  three  parts. 
T.     What  is  each  part? 
C.     One-third. 


FRACTIONS.  71 

T.      Into  how  many   parts  is  the  line   divided    by   the   short 
lines  ? 


C.      Into  six  parts. 

T.     What  is  each  part? 

C.     One-sixth. 

T.      How  many  one-sixths  in  one-third? 

C.     There  are  two  one-sixths  in  one-third. 

T.      How  many  one-sixths  in  one-half? 

C.     There  are  three  one-sixths  in  one-half. 

T.     How  many  times  is  one-sixth  contained  in  one-third? 

C.     One-sixth  is  contained  in  one-third  twice. 

T.     How  many  times  is  one-sixth  contained  in  one-half? 

C.     One-sixth  is  contained  in  one-half  three  times. 

T.      How  many  times  is  two-sixths  contained  in  three-sixths? 

C.  Two-sixths  is  contained  in  three-sixths  one  and  one-half 
times. 

T.     How  many  times  is  one-third  contained  in  three-sixths? 

C.  One-third  is  contained  in  three-sixths  one  and  one-half 
times. 

T.     Then  how  many  times  is  one-third  contained  in  one-half? 

C.     One-third  is  contained  in  one-half  one  and  one-half  times. 

T.     How  much  is  one-half  times  three? 

C.  One-half  times  three  equals  three-halves,  or  one  and  one- 
half. 

T.     How  much  is  one-half  divided  by  one-third? 

C.     One-half  divided   by  one-third  equals  one  and   one-half. 

T.     What  is  one-half  multiplied  by  three? 

C.  One-half  multiplied  by  three  equals  three-halves,  or  one 
and  one-half. 

T.  Then  one-half  divided  by  one-third  equals  one-half  mul- 
tiplied by  three,  does  it? 

C.     It  does. 

T.     What  is  the  difference  between  the  two  expressions? 

C.  The  latter  has  the  three  above  instead  of  below  the  line, 
and  is  multiplication  instead  of  division. 

T.  Let  us  see  whether  we  may  always  do  as  we  have  done 
in  this  example. 


72  THK  S  UP  PL  Y. 

T.      Ho\v  much  is  two-thirds  divided  by  one-half? 

C.     Two-thirds  divided  by  one-half  equals  one  and  one-third. 

T.     What  is  two-thirds  multiplied  by  two? 

C.     Two-thirds  multiplied  by  two  equals  one  and  one-third. 

T.     What  do  you  see  this  time? 

C.  I  see  that  two-thirds  divided  by  one-half  equals  two- 
thirds  multiplied  by  two. 

T.  Well,  you  may  try  other  examples  and  see  whether  you 
can  find  any  in  which  this  is  not  true. 

This  comparison  of  results  should  be  continued  as  other 
fractional  forms  are  taught  until  sufficient  experience  is  gained 
by  the  pupil  to  cause  him  to  be  certain  that  questions  of  that 
kind  can  be  solved  in  either  manner,  when  it  may  be  given  him 
as  a  rule  that,  In  dividing  one  fraction  by  another  the  divisor 
may  be  inverted  and  the  methods  of  multiplication  employed. 

It  should  also  be  drawn  out  what  the  object  of  so  doing  is; 
to-wit,  convenience. 

Sevenths  are  developed  as  were  fifths. 

Eighths  are  developed  as  were  sixths,  halves  and  fourths  both 
being  again  considered,  but  this  time  with  eighths  as  units  of 
measure. 

Ninths  are  developed  in  the  same  manner  as  fourths  were, 
thirds  being  considered  with  them. 

The  following  will  indicate  the  character  of  the  co-ordinate 
work  at  this  time: 

21)210(  £=  i  +  i  =  \~\^  2xf= 

16)320(  |=  l+f  =  f-l=  3x1  = 

42)1260(  |«  1+1=  1-1  =  4x1  = 

13)86(  f=  1+1-  |-1=  5x1= 

35)99(  f=  1+1=  1-1=  3x1  = 

41)128(  f=  |+1=  1-1=  4x1= 

93)325(  f=  1+1=  1-1=  6xf  = 

103)842(  f=  l+f=  2-1=  9x1  = 

82)672(  ft=  |+1=  2-1= 

75)567(  1+1=  2-1= 

91)635(  1+1=  11-1=  8x|  = 

107)932(  1+2=  11-1=  7xf= 


4-5-2-  x2 


FRACTIONS. 


|  of  2  = 
2-5-2  — 

of  1 


i-f 


1.1 
?~¥ 

1x3 


All  work  in  fractions  should  be  diagramed  by  the  pupils  until 
they  are  perfectly  clear  in  their  comprehension  of  the  principles 
of  fractions;  then  the  use  of  diagrams  should  be  discontinued. 

The  following  will  indicate  the  character  of  the  diagram  that 
will  produce  satisfactory  results: 


-s-^=f.     Same  as  above. 

xf=f.     Should  not  be  diagramed  because  of  its  simplicity. 

of  2 -4. 


The  first  diagram  will  also  illustrate  questions  like  the  follow 
ing: 

How  many  times  is  one-fourth  contained  in  one- half? 
How  many  times  is  one-half  contained  in  one-fourth? 
How  many  times  is  one-half  contained  in  three-fourths? 


74  THE  SUPPL  Y. 

How  many  times  is  three-fourths  contained  in  one-half? 
The  second  diagram  will  also  illustrate  questions  like  the  fol- 
lowing: 

How  many  are  one-half  and  one-third? 

How  many  are  one-half  and  one-sixth? 

What  is  the  difference  between  one-half  and  one-sixth  ? 

What  is  the  difference  between  two-thirds  and  one-half? 

What  is  one-half  of  one-third? 

What  is  one-third  of  one-half? 

What  is  two-thirds  of  one-half? 

How  many  times  is  one-sixth  contained  in  one-half? 

How  many  times  is  one-third  contained  in  one-half? 

How  many  times  is  one-half  contained  in  one-third? 

How  many  times  is  two-thirds  contained  in  one-half? 

How  many  times  is  five-sixths  contained  in  two-thirds? 


FRA  CTIONS--DECIMA  LH. 


FRACTIONS— DECIMALS 


The  subject  of  tenths  introduces  to  the  pupils  new  percepts  for 
the  same  concepts  under  given  circumstances.  The  manner  of 
treating  tenths  is  precisely  the  same  as  that  of  treating  any  other 
fractional  concept,  except  that  when  the  name  is  drawn  from  the 
pupil  and  its  numerical  equivalent  written,  it  is  to  be  written  first 
as  ^  and  secondly  as  .1.  The  manner  of  proceeding  may  be 
as  follows: 

T.  (Dividing  a  line  into  ten  equal  parts.)  Into  how  many 
parts  is  this  line  divided? 

C.     Into  ten  equal  parts. 

T.     What  is  each  part? 

C.     Each  part  is  one-tenth. 

T.     You  may  write  one-tenth  on  your  slates. 

c.    iV 

T.  Very  well;  I  will  show  you  another  way  to  write  one- 
tenth:  (.1)  one  with  a  period  before  it. 

T.     You  may  now  write  two-tenths  in  both  ways. 

T.     You  may  now  write  one  and  one-tenth  in  both  ways. 

T.     You  may  now  write  three  and  one-tenth. 

T.     You  may  now  write  twenty-one  and  one-tenth. 

As  one-tenth  will  in  practice  be  used  more  frequently  in  the 
decimal  form,  it  should  be  given  most  attention  as  a  decimal  form. 
It  should  be  compared  in  value  with  units,  as  units  with  tens  and 
tens  with  hundreds,  to  clearly  show  the  pupil  that  it  may  be 
treated  in  all  respects  as  another  digit.  To  that  end  many  ex- 
ercises like  the  following  should  be  given: 


THE  SUPPLY. 


2X 


.5 
.4 
4 

40 

11 

1.1 

9 

.9 

90 

7 

w 

0 
.0 


3x 


7 

80 
3 
.3 

12 

1.2 

8 

80 

.8 

5 

.5 

9 

.9 


.4x 


3 

30 
6 

60 

8 

80 

70 

7 

0 

00 

9 

90 

1 

10 


.t  X 


5 
50 

2 

20 

7 

70 

4 

40 

3 

30 

60 

6 

80 

8 


.4 

4 

40 

.7 

7 

70 

.1 

1 

10 

90 

9 

.9 

.8 

8 


4 
.4 
1 
.1 

8 
.8 


1.2- 


1.: 


6 

.6 

4- 

.4 

3 

.3 

2 

.2 


1.8-4- 


Addition: 


84.5 
35.2 
81.9 
73.6 
48.4 


Subtraction :    83. 5 
72.9 


Division:    5)86.5 


Multiplication:   86.4 
28 


85)95.3( 


The  further  consideration  of  fractional  forms  may  now  be  pro- 
ceeded with  without  the  use  of  either  objects  or  lines,  making  use 
of  them  only  in  rare  cases  if  at  all. 

The  subject  of  hundredths  may  profitably  be  considered  while 
many  other  fractional  forms  between  tenths  and  hundredths  have 
received  no  attention  whatever.  The  reasons  for  this  will  be 
apparent,  when  it  is  considered  that'  hundredths  are  in  constant 
use,  while  nineteenths,  twenty-thirds,  and  many  other  fractional 
units  are  rarely  used  at  all. 


FJL  t  crroys—  D  AY  'IMA  LS.  n 

The  pupil  is  now  in  the  very  center  of  supply  and  has  only  to 
be  educated  in  the  subject  of  demand  to  make  him  an  adept  in 
the  ordinary  business  affairs  of  life.  The  ability  to  select  with 
promptness,  to  utilize  with  ease,  to  know  his  needs,  to  husband 
his  resources;  these  are  yet-to-be-acquired  forces  that  can  only 
be  the  outgrowth  of  actual  affairs  or  of  wise  leading  on  the  part 
of  a  thoroughly  skilled  teacher  who  is  able  to  send  into,  and 
draw  from,  the  business  world  what  is  needed,  using  as  his 
agents  the  very  material  into  whose  hands  he  wishes  to  place 
these  new  forces. 

The  matter  of  pointing  off  in  multiplication  and  division  of 
decimals  needs  to  be  dealt  with  rigidly  at  this  time,  so  that  the 
applications  of  percentage  will  not  abound  in  difficulties  not 
belonging  to  percentage  at  all  but  to  the  subject  of  decimals 
altogether.  A  few  illustrations  by  means  of  changing  from  the 
decimal  to  the  common  form  and  care  thereafter  will  suffice. 


78  THE  SUPPLY. 


DECIMALS— PER   CENT. 


When  the  subject  of  decimals  is  thoroughly  understood,  the 
idea  of  per  cent  may  be  added  with  little  difficulty.  The  subject 
of  hundredths  should  be  reviewed  and  the  subject  of  per  cent 
introduced  in  some  such  way  as  this: 

T.     What  have  I  written?     (.02.) 

C.     Two-hundredths. 

T.  It  is  also  called  two  per  cent;  per  cent  meaning  hun- 
dredths. 

T.     You  may  write  three  per  cent. 

C.     .03. 

T.  Four  per  cent;  five  per  cent:  twelve  per  cent;  eighty-five 
per  cent;  one-hundred-eighteen  per  cent. 

C.     .04;    .05;    .12;    .85;    1.18. 

T.  These  numbers  are  written  in  another  way  also:  4%;  5%; 
12%;  85%;  118%. 

T.  You  may  write  each  of  these  in  three  ways,  and  notice  the 
change  in  the  location  of  the  decimal  point. 


12%  ==.12=^  or  ^. 

85%  =.85=jfo  or  tf. 
118%  =1.18  =^=1^  brl-fo. 

T.     Write  one-tenth  per  cent  in  like  manner. 
C.     .1%=.001=T^. 

T.     How  about  the  location  of  the  decimal  point? 
C.     The  decimal  point  is  two  places  farther  to  the  left  in  the 
decimal  form  than  it  is  in  the  per  cent  form. 
T.     This  is  always  so. 


DECIMALS— PER  CEXT  79 

Much  practice  in  the  writing  of  equivalents  is  an  essential 
preface  to  a  ready  application  of  the  principles  of  percentage. 
The  order  of  the  columns  should  be  changed  so  that  perfect 
facility  in  the  writing  of  equivalents  may  be  assured.  The 
columns  should  also  be  written  singly  and  read  in  the  different 
ways,  until  no  hesitation  whatever  is  to  be  noticed.  Then  much 
oral  work  of  the  following  nature  should  be  given. 

What  is  one  one-hundredth  of  one  hundred? 

What  is  one  per  cent  of  one  hundred? 

What  is  two  one-hundredths  of  one  hundred? 

What  is  two  per  cent  of  one  hundred? 

What  is  two  one-hundredths  of  two  hundred? 

What  is  two  per  cent  of  two  hundred? 

What  is  three  per  cent  of  two  hundred? 

What  is  four  per  cent  of  two  hundred? 

What  is  five  per  cent  of  two  hundred? 

What  is  five  per  cent  of  three  hundred? 

What  is  six  per  cent  of  three  hundred? 

One  is  one  per  cent  of  what  number? 

Two  is  one  per  cent  of  what  number? 

Two  is  two  per  cent  of  what  number? 

Short  and  clear  explanations  should  accompany  all  work  in 
percentage.  The  following  may  be  considered  as  fair;  though 
care  must  be  exercised  that  the  pupils  do  not  commit  a  form  to 
memory  without  having"  grasped  its  meaning: 

What  is  five  per  cent  of  three  hundred? 

One  per  cent  of  three  hundred  is  three,  and  five  per  cent 
of  three  hundred  is  five  times  three,  or  fifteen.  Or 

Five  per  cent  equals  one-twentieth;  and  one-twentieth  of  three 
hundred  is  fifteen. 

Two  is  two  per  cent  of  what  number? 

Since  two  is  two  per  cent  of  a  number,  one  per  cent  of  the 
same  number  is  one,  and  one  hundred  per  cent  of  the  number  is 
one  hundred  times  one,  or  one  hundred.  Or 

Since  two  is  two  per  cent,  or  one-fiftieth,  of  a  number,  fifty- 
fiftieths  of  the  same  number  is  fifty  times  two,  or  one  hundred. 

Two  is  what  per  cent  of  five? 

Two  is  two-fifths  of  five;  two-fifths  of  five  equals  forty  hun- 
dredths,  or  forty  per  cent,  of  five;  hence  two  is  forty  per  cent  of 
five. 


SO  THE  SUPPLY. 

After  a  clear  idea  of  per  cent  is  gained,  the  tables  ol  compound 
numbers  should  be  reviewed,  formulated,  and  completed,  so  that 
the  business  applications  may  present  as  few  difficulties  as  pos- 
sible. A  general  review  of  such  points  as  were  troublesome,  and 
of  points  whose  especial  needs  are  recognized,  should  also  be 
made  now  and  as  frequently  as  seems  necessary.  Frequent 
reviews  of  carefully  selected  points  should  be  considered  not 
only  as  proper  but  as  absolutely  indispensable. 


TABULTAIOXS. 


TABULATIONS. 


"Exercise  involves  repetition,  which,  as  regards  bodily  actions, 
ends  in  habits  of  action  and,  as  regards  impressions  received  by 
the  mind,  ends  in  clearness  of  perception." — Payne. 


In  developing  the  ideas  of  the  simple  numbers  the  disjointed 
facts  of  the  tables  are  incidentally  developed,  so  -that  it  only 
remains  to  systematize  these  facts  and  introduce  applications  in 
which  the  ideas  of  trade  are  the  central  ones. 

MONEY. 

T.     How  many  cents  in  a  dime  ? 

C.     There  are  ten  cents  in  a  dime. 

T.      How  many  dimes  in  a  dollar? 

C.     There  are  ten  dimes  in  a  dollar. 

T.  One-tenth  of  a  cent  is  called  a  mill;  then  how  many  mills 
are  there  in  a  cent  ? 

C.     There  are  ten  mills  in  a  cent. 

T.  I  will  now  put  these  facts  together  so  that  you  may  see 
the  table  which  you  are  to  remember: 

10  mills   =1  cent. 
10  cents  =1  dime. 
10  dimes  — 1  dollar. 

The  term  "eagle,"  not  being  upon  any  of  the  coins  and  rarely 
used,  should  not  be  inserted  in  the  table.  The  mill,  though  not  a 
coin,  is  in  constant  use  in  the  reading  of  numbers  and  in  com- 
putations, and  hence  should  be  inserted.  Cent,  dime,  and 
dollar  are  upon  the  coins  themselves. 


si-  THE  8V PPL  Y. 

CAPACITY. 

T.  How  many  pints  in  a  quart? 

C.  There  are  two  pints  in  a  quart. 

T.  How  many  quarts  in  a  gallon  ? 

C.  There  are  four  quarts  in  a  gallon. 

T.  How  many  gallons  in  a  barrel? 

C.  There  are  thirty-one  and  one-half  gallons  in  a  barrel, 

T.  How  many  barrels  in  a  hogshead? 

C.  There  are  two  barrels  in  a  hogshead. 

T.  These  facts  I  will  write  for  you  in  a  table, 

LIQUID  MEASURE. 

2  pints     =1  quart. 
4  quarts  =1  gallon, 
#l-j  gallons  =  1  barrel. 
2  barrels--!  hogshead, 

T.  Anna,  you  may  fetch  the  pint  measure,'  Jennie,  you  may 
fetch  the  quart  measure;  and  Hattie,  you  may  fetch  the  gallon 
measure, 

T.     Class,  what  do  people  measure  with  these  measures  ? 

C.  Water,  sirup,  milk,  vinegar,  wine,  honey,  beer,  and  all 
other  liquids, 

T.     With  what  do  people  measure  wheat? 

C.  People  measure  wheat  with  peek,  half-bushel,  and  bushel 
measures. 

T.      How  many  quarts  in  a  peck  ? 

C.     There  are  eight  quarts  in  a  peck, 

T.      How  many  pecks  in  a  bushel? 

C.     There  are  four  pecks  in  a  bushel. 

T.     I  will  write  these  facts  also  in  a  table, 

DRY  MEASURE. 

2  pints  =1  quart. 
8  quarts  — 1  peck. 
4  pecks  —1  bushel. 

T,     Lily,  you  may  fetch  the  quart  measure  J  Charles,  you  may 


TABULATIONS.  88 

fetch  the  peck  measure;  and  Samuel,  you  may  fetch  the  bushel 
measure. 

T.  Lily,  is  your  quart  measure  of  the  same  capacity  as  the 
one  that  Jennie  brought  us? 

L.  My  quart  measure  is  somewhat  larger  than  the  one  Jennie 
brought. 

T.     What  do  people  measure  with  these  measures  ? 

C.  Grain,  vegetables,  and  all  other  dry  or  bulky  articles  that 
are  bought  and  sold  by  the  quart,  peck,  or  bushel. 

T.  Are  these  measures  used  as  generally  on  the  Pacific  coast 
as  on  the  Atlantic  coast? 

C.  No;  grain,  vegetables,  etc.,  are  generally  bought  and  sold 
by  weight  on  the  Pacific  coast. 

WEIGHT. 

T.  How  many  ounces  in  a  pound  ? 

C.  There  are  sixteen  ounces  in  a  pound. 

T.  How  many  pounds  in  a  hundred-weight? 

C.  There  are  one  hundred  pounds  in  a  hundred-weight. 

T.  How  many  hundred-weight  in  a  ton? 

C.  There  are  twenty  hundred-weight  in  a  ton. 

T.  These  facts  may  now  be  written: 

AVOIRDUPOIS   WEIGHT. 

16  ounces  =1  pound. 

100  pounds  =1  hundred- weight. 

20  hundred-weight  =  1  ton. 

T.  Lillian,  you  may  fetch  the  ounce  weight;  and  Clara,  you 
may  fetch  the  pound  weight. 

T.     What  articles  are  bought  and  sold  with  these  weights  ? 

C.  Sugar,  meat,  hay,  grain,  live-stock,  lime,  flour,  iron, 
copper,  coal,  and  all  other  ordinary  articles  of  merchandise  that 
are  bought  and  sold  by  weight. 

T.  Do  we  use  these  weights  in  buying  and  selling  gold  or 
silver  ? 

C.     We  do  not. 

T.  Levi,  you  may  fetch  me  the  ounce  weight  with  which 
people  weigh  gold  and  silver. 


84  THE  SUPPLY. 

T.  Is  this  ounce  weight  heavier  or  lighter,  Lillian,  than  the 
ounce  weight  you  have? 

L.     It  is  heavier. 

T.  How  many  of  the  ounces  that  Levi  has  make  a  pound 
with  which  to  weigh  gold  and  silver? 

C.     Twelve  ounces. 

T.  Cora,  you  may  fetch  the  pound  weight  of  which  we  are 
now  talking. 

T.  Does  this  pound  weight  weigh  more,  or  less,  than  the 
pound  weight  with  which  we  weigh  sugar? 

C.      It  weighs  less, 

T.  Then  the  ounce  with  which  we  weigh  gold  and  silver 
weighs  more,  and  the  pound  weighs  less,  does  it,  than  the  cor- 
responding weight  with  which  we  weigh  sugar  and  coffee? 

C.     It  does, 

T.     The  table  is: 

TROY  WEIGHT. 

24  grains  =1  pennyweight, 

20  pennyweights  — 1  ounce. 
12  ounces  — 1  pound, 

T.  These  weights  are  used  in  weighing  gold,  silver,  platinum, 
and  all  jewels  except  diamonds  and  pearls, 

T.  After  school  you  may  go  with  me  to  the  drug  store  of 
Deveny  &  Crew,  and  we  will  learn  whether  they  use  any  of  these 
weights,  and  whether  they  also  use  others  of  which  we  have 
none. 

This  excursion  may  be  made  of  much  interest  as  well  as 
pleasure,  as  all  novel  methods  of  procedure  excite  at  least  tem- 
porary interest,  and,  if  well  planned  and  systematically  carried 
out,  the  lessons  learned  should  be  of  the  most  durable  character. 
These  excursions  being  after  school,  the  pupils  should  be  as  free 
from  restraint  as  possible;  in  fact,  if  the  teacher  is  th,e  leader  he 
or  she  should  be,  no  act  of  any  pupil  is  likely  to  occur  that  Will, 
in  any  way,  mar  the  pleasure  of  the  occasion.  These  little  outings 
may  be  frequently  indulged  in,  if  there  be  a  definite  object  in 
view,  and  will  therefore  be  a  profitable  as  well  as  pleasant  feature 
of  the  school.  The  interview  with  Deveny  &  Crew  should  be  as 
informal  as  possible.  These  gentlemen  will  take  great  pleasure 


TABULATIONS.  so 

(business  men  always  do)  in  telling  the  pupils  all  about  the 
weighing,  in  showing  them  the  weights,  and  in  weighing  for 
them  different  articles  and  comparing  their  weights.  The  in- 
formation thus  obtained  will  need  to  be  formulated  the  next  day 
in  the  classroom,  but  the  impression  made  will  be  lasting. 

T.  (The  day  following  the  excursion.)  Gertrude,  you  may 
tell  what  you  learned  about  weight  from  Deveny  &  Crew  last 
evening. 

G.  I  learned  that  in  mixing,  or  compounding,  medicines  they 
use  weights  that  resemble  the  Troy  weights  somewhat.  Their 
grain,  ounce,  and  pound  are  exactly  the  same  as  the  grain, 
ounce,  and  pound  of  the  Troy  weight.  They  use  the  scruple 
and  the  dram  instead  of  the  pennyweight.  Their  table  is  as 
follows; 

APOTHECARIES'   WEIGHT. 

20  grains     =1  scruple. 

8  scruples  — 1  dram. 

S  drams     =1  ounce. 
12  ounces    —1   pound. 

T.  That  is  right.  Ralph,  you  may  tell  me  what  else  you 
learned  about  weight  last  evening. 

R.  I  learned  also  that  they  sell  by  Avoirdupois  weight  such 
goods  as  are  sold  by  weight,  and  that  for  liquids  they  have 
measures  of  capacity  different  from  any  we  have  ever  talked 
about  in  class, 

T.     You  may  tell  us  what  you  learned  about  them. 

R.  I  learned  that  the  table  differs  from  the  ordinary  liquid 
measure  in  having  subdivisions,  of  the  pint.  The  table  is: 

APOTHECARIES'  LIQUID   MEASURE. 

tiO  minims  =  1  dram. 
<S  drams   =1   ounce. 
K>  ounces  —  1  pint. 

And  the  measures  are  graduated  glass  vessels  in  which  very 
small  quantities  may  be  accurately  measured.  These  measures 
are  used  both  for  mixing,  or  compounding,  medicines  and  in 
detailing  the  same  when  in  liquid  form. 


86  THE  SUPPLY. 

The  following  will  indicate  the  character  of  the  work  necessary 
to  impress  the  tables  upon  the  pupils'  minds: 

Reduce  twenty-five  pounds  and  eight  ounces,  Avoirdupois 
weight,  to  the  decimal  of  a  hundred-weight. 

In  forty  pounds  and  eight  ounces,  Avoirdupois  weight,  how 
many  pounds,  Troy  weight? 

At  one  dollar  and  thirty  cents  per  cental,  how  many  tons  of 
wheat  can  I  buy  for  two  thousand  six  hundred  dollars? 

I  bought  675  lb.  of  hay  at  $14  per  ton ;    what  did  it  cost  me  ? 

I  have  47  Ib.  of  silver,  Avoirdupois  weight;  what  is  it  worth  at 
SI  per  ounce  Troy?  At  two  per  cent  more  per  ounce? 

TIME. 

T.  How  many  seconds  in  a  minute? 

C.  There  are  sixty  seconds  in  a  minute. 

T.  You   may   take  my  watch,  Fred,  and  show  the  class  the 

time  measured  by  a  minute. 

T.  I  will  say  "one"  at  the  beginning  of  a  minute  and  "one" 

again  at  its  end. 

T.  How  many  minutes  in  an  hour? 

C.  There  are  sixty  minutes  in  an  hour. 

T.  How  many  hours  in  a  day? 

C.  There  are  twenty-four  hours  in  a  day. 

T.  What  is  a  day  ? 

*C.  A  day  is  the  time  required  for  the  earth  to  revolve  once 

upon  its  axis. 

T.  When  does  the  day  begin  ? 

C.  At  midnight. 

T.  When  does  it  end? 

C.  At  the  next  midnight. 

T.  How  many  days  in  a  week  ? 

C.  There  are  seven  days  in  a  week. 

T.  How  many  days  in  an  ordinary  year? 

C.  Three  hundred  sixty-five  days. 

T.  How  many  days  in  a  leap  year? 

C.  Three  hundred  sixty-six  days. 

T.  What  is  a  year? 

*This  answer  is  not  strictly  correct  hut  the  error  cannot  be  pointed  out  at  this 
time. 


C.  A  year,  is  the  time  required  for  the  earth  to  revolve 
around  the  sun  once. 

T.  Yes;  the  earth  revolves  around  the  sun  in  365  days,  5 
hours,  48  minutes,  and  4(5.4  seconds.  For  convenience,  365 
days  are  called  a  year.  This  leaves  an  excess  of  5  hours,  4S 
minutes,  and  46.4  seconds,  arid,  in  four  years,  an  excess  of  four 
times  5  hours,  48  minutes,  and  46.4  seconds,  or  an  excess  of  23 
hours,  15  minutes,  and  5.0  seconds,  very  nearly  a  day;  there- 
fore ordinarily  each  year  divisible  by  four  is  given  366  days. 
This  however  causes  a  deficiency  of  44  minutes  and  54.4 
seconds,  and  in  a  century,  or  twenty-five  times  four  years,  it 
causes  a  deficiency  of  twenty-five  times  44  minutes  and  54.4 
seconds,  or  a  deficiency  of  18  hours,  42  minutes,  and  40  seconds; 
hence  the  century  year  ordinarily  is  given  only  305  days.  This 
then  produces  an  excess  of  5  hours,  1 7  minutes,  and  20  seconds 
every  century,  or  an  excess  of  four  times  5  hours,  17  minutes, 
and  20  seconds,  or  an  excess  of  21  hours,  1)  minutes,  and  20 
seconds,  in  four  centuries;  hence  every  century  year  divisible  by 
400  is  given  360  days.  This  again  produces  a  deficiency  of  2 
hours,  50  minutes,  and  40  seconds,  and  in  four  thousand  years, 
or  ten  times  four  hundred  years,  it  produces  a  deficiency  of  ten 
times  2  hours,  50  minutes,  and  40  seconds,  or  a  deficiency  of  28 
hours,  26  minutes,  and  40  seconds:  hence  4000  will  probably  be 
given  only  365  days.  Thus  leap  years  are  years,  except  century- 
years,  divisible  by  four,  and  century  years,  except  multiples  of 
4000,  divisible  by  400. 

TIME  TABLE. 

<>0  seconds  — 1  minute. 
60  minutes  =  1  hour. 
24  hours     —1   day. 

7  days       =1  week. 
30  days       —1  business  month. 
12  months  =1  year. 

365  days        =1   year. 

366  days       —  1  leap  year. 

TESTS. 

Reduce  7  weeks,  4  days,  and  4  hours,  to  the  fraction  of  a  year. 
To  the  decimal  of  a  year.  What  per  cent  of  a  year? 


ss  niK  fir P PLY. 

What  time  elapsed  from  January  1,  1850,  to  July  1(>,  1872? 

How  many  days  are  there  in  each  of  the  months? 

How  many  days  are  there  from  January  14  to  May  2H? 

How  many  days  are  reckoned  as  a  month,  in  computing 
interest? 

How  many  weeks  are  there  in  a  year? 

Six  months  are  what  per  cent  of  a  year?  Three  months  ?  Four 
months?  Two  months?  Eight  months?  Nine  months?  Ten 
months  ?  One  month  ? 

DISTANCE. 

T.     How  many  inches  in  a  foot  ? 

C.     There  are  twelve  inches  in  a  foot. 

T.  John,  you  may  draw  a  line  upon  the  blackboard  one  foot 
in  length,  and  divide  it  into  inches. 

T.  You  may  now  take  the  foot-rule  and  see  how  accurately 
you  have  done  your  work. 

T.     How  many  feet  in  a  yard  ? 

C.     There  are  three  feet  in  a  yard. 

T.  Walter,  you  may  draw  a  line  one  yard  in  length,  and 
divide  it  into  feet. 

T.     You  may  now  test  your  work  with  the  foot-rule. 

T.      How  many  yards  in  a  rod  ? 

C.     There  are  five  and  one-half  yards  in  a  rod. 

T.     How  many  rods  in  a  mile? 

C.     There  are  three  hundred  and  twenty  rods  in  a  mile. 

T.  You  may  now  each  think  of  two  houses  or  two  places  or 
two  objects  of  any  kind  that  are  one  mile  apart. 

TABLE  OF  DISTANCES. 

VI  inches =1  foot. 
3  feet      =1   yard. 
5|  yards  =1  rod. 
320  rods    =1  mile. 

T.  George,  I  shall  write  a  note  to  County  Surveyor  Atherton, 
requesting  him  to  lend  us  his  Surveyors'  chain  at  such  time  as 
shall  be  most  convenient  for  him,  and  I  will  thank  you  to  deliver 


TABULATIONS.  ^ 

the  note  immediately  after  school,  as  County  officers  close  their 
offices  at  five  o'clock. 

T.  (On  the  following  day,  George  having  brought  the  Sur- 
veyors' chain.)  Benjamin,  you  may  measure  the  length  of  this 
chain. 

B.     It  is  exactly  66  feet  long. 

T.  You  may  now  measure  the  distance  from  one  small  link 
to  the  corresponding  next  small  link. 

B.  It  lacks  a  little  of  being  eight  inches. 

T.  Yes;  it  is  exactly  7.92  inches  and  that  distance  is  called  a 

link.  There  are  one  hundred  of  these  in  the  chain. 

T.  66  feet  equal  how  many  rods? 

C.  66  feet  equal  4  rods. 

T.     This  chain  is  used  by  land  surveyors. 

SURVEYORS'    MEASURE. 

7.92  inches  =  1  link. 

25  links          =1  rod. 

4  rods          =1  chain. 

80  chains       =1  mile. 

TESTS. 

Reduce  84,683  inches  to  integral  higher  denominations. 

From  4  miles,  subtract  2  miles,  25  rods,  2  yards,  2  feet,  and  9 
inches,  and  find  twenty-five  per  cent  of  the  remainder. 

Divide  85  miles,  73  rods,  4  yards,  2  feet,  and  8  inches  into  18 
equal  parts. 

Reduce  3  miles,  2  rods,  4  yards,  and  8  inches  to  inches. 
To  links. 

A  and  B  start  at  the  same  time  to  run  toward  each  other,  when 
they  are  one  mile  apart;  A  runs  three-fourths  as  fast  as  B;  how 
far  does  each  run  before  they  meet? 

A  room  is  40  feet  long  and  35  feet  wide;  what  will  be  the 
expense  of  carpeting  it  with  carpet  f  of  a  yard  wide,  the  strips 
running  lengthwise  of  the  room,  there  being  no  loss  in  matching, 
and  the  price  being  $2. 25  per  yard  ? 

How  many  inches  higher  is  a  horse  that  measures  16^  hands 
than  one  that  measures  14f  hands  ? 


JM»  THE  SUPPLY. 

How  many  boards  of  the  longest  possible  equal  lengths  will 
inclose  a  rectangular  field,  9,893  feet  long  and  8,047  feet  wide,- 
with  a  straight  fence,  six  boards  high? 

What  will  it  cost,  at  two  dollars  per  yard,  to  carpet  a  room,  20 
feet  long  and  18  feet  wide,  with  carpet  three-fourths  of  a  yard 
wide,  the  design  being  one  yard  in  length,  and  the.  strips  to  run 
so  as  to  make  as  little  expense  as  possible. 

SQUARE  MEASURE. 

The  table  of  distances,  or  Linear  measure,  having  been  thor- 
oughly taught  in  connection  with  the  simple  numbers,  the  ideas 
of  squares  and  square  units  should  be  systematically  presented  in 
some  such  way  as  this: 

T.     James,  you  may  draw  a  square  upon  the  blackboard, 

J- 


T.     Samuel,  you  may  draw  a  line  one  foot  long. 
S.  

T.     James,  you  may  use  Samuel's  foot-line  and  upon  it,  as  one 
side,  draw  a  square. 


T.  Samuel,  you  may  tell  me  the  name  of  this  square. 

S.  It  is  a  square  foot. 

T.  Then  what  is  a  square  foot,  Mary? 

M,  A  square  foot  is  a  square  each  of  whose  sides  is  one  foot 
long. 


TABULTAION8. 


T.     Minnie,  you  may  divide  each  of  the  sides  into  inches, 
using  the  foot-rule. 

M. 


T.     How  many  inches  in  a  foot? 
C.     There  are  12  inches  in  a  foot. 

T.     Alice,  you  may  draw  parallel  lines  connecting  the  points 
on  the  opposite  sides. 

A. 


T. 
C. 
T. 
C. 
T. 
C. 
T. 


Class,  what  kind  of  squares  are  these  small  squares  ? 
They  are  square  inches. 

How  many  square  inches  are  there  in  each  row  ? 
There  are  12  square  inches  in  each  row. 
How  many  rows  are  there  ? 
There  are  12  rows. 

Then   how   many   square   inches  are  there  in    the  large 
square  ? 

C.     There  are  12  times  12  square  inches,  or  144  square  inches, 
in  the  large  square. 

T.     What  is  the  large  square? 
C.     It  is  a  square  foot. 

T.     Then  how  many  square  inches  are  there  in  one  square 
foot? 


THE  8  UP  PL  Y. 


C.  There  are  144  square  inches  in  one  square  foot. 

T.  I  will  now  write  this  where  we  may  see  it  all  the  time, 

T.  Herbert,  you  may  draw  a  line  one  yard  in  length. 

H. 

T.  How  many  feet  in  a  yard? 

C.  There  are  3  feet  in  one  yard, 

T.  Susan,  you  may  draw  a  square  upon  Herbert's  line. 

S, 


T.  What  is  the  name  of  this,  square  ? 

S.  A  square  yard. 

T.  Mortimer,  you  may  divide  each  side  into  feet, 

M. 


T.     Lizzie,  you  may  draw  lines  parallel  to  the  sides  connect- 
ing these  points  of  division. 

L, 


TABULATIONS 


1J3 


T.  What  is  each  of  these  smaller  squares? 

C.  A  square  foot. 

T.  How  many  smaller  squares  in  the  large  square  ? 

C.  There  are  nine  smaller  squares  in  the  large  square. 

T  Then  how  many  square  feet  are  there  in  one  square  yard  ? 

C.  There  are  9  square  feet  in  one  square  yard. 

T.  I  will  place  this  fact  under  the  fact  we  learned  a  little 

while  ago. 

T.  How  many  yards  in  a  rod  ? 

C.  There  are  5^-  yards  in  a  rod. 

T.  Mabel,  you  may  draw  a  line  one  rod  in  length. 


M. 


T.     You  may  now  draw  a  square  upon  this  line  as  one  side. 

M.    The  board  is  not  large  enough. 

T.     What  shall  we  do  then  ? 

M.     We  may  use  a  shorter  line  and  call  it  a  rod. 

T.  Yes;  you  may  use  two  inches  for  a  yard.  Then  how 
many  two-inches  will  you  need  for  a  rad? 

M.  I  shall  need  5  two-inches  and  1  one-inch,  or  eleven 
inches. 

T.  That  is  right;  you  may  now  draw  your  line  which  we  are 
to  use  as  a  rod. 


M. 


T.     Kate,  you  may  draw  a  square  upon  this  line  as  one  side, 
divide  the  sides  into  yards,  and  draw  the  parallel  lines. 


K. 


T.     Are  these  divisions  all  squares? 

C.     No;  those  on  the  right  and  those  at  the  bottom  are  half- 
squares. 


THE  SUPPLY. 


T.  What  is  the  one  in  the  corner? 

C.  It  is  the  half  of  a  half-square,  or  a  fourth  of  a  square. 

T.  That  is  right;  let  us  see  how  many  square  yards  we  have. 

C.  We  have  25  square  yards. 

T.  How  many  half  square  yards? 

C.  JO  half  square  yards,  or  five  square  yards  more. 

T.  Then  how  many  square  yards  are  there  in  the  square  rod  ? 

C.  There  are  25  square  yards  and  5  square  yards  and  \  of  a 
square  yard,  or  30^  square  yards. 

T.  We  will  now  write  this  also  with  our  other  facts  and  com- 
plete the  table: 

SQUARE  MEASURE. 


144  square  inches 
9  square  feet      ; 
30^  square  yards 
160  square  rods 


640  acres 


square  foot. 

square  yard. 
=  1  square  rod. 
—  1  acre. 
— 1  square  mile,  or  1  section. 


T.     Avis,  you  may  draw  a  figure  3  inches  long  and  2  inches 
wide,  divide  its  sides  into  inches,  and  draw  parallel  lines. 

A. 


T.  How  many  square  inches  in  a  rectangle  3  inches  long 
and  2  inches  wide? 

A.     There  are  three  times  two,  or  6,  square  inches. 

Similar  simple  problems  may  be  given  until  the  pupils  are 
able  to  state  the  process  for  solving  problems  of  that  character. 
Blackboard  forms  of  solution,  accompanied  by  good  explanations, 
should  follow,  until  the  subject  is  firmly  fixed  in  the  minds  of  the 
pupils: 

Problem:  Measure  this  room  and  determine  how  many  square 
feet  there  are  in  the  floor. 


Solution; 


TABULATIONS. 

45  square  feet. 
36 


225 
135 
1575  square  feet. 

45  feet. 


I .")".">  S(|ll;tlV  !<><•!. 


Explanation:  A  surface  1  foot  long  and  1  foot  wide  contains 
1  square  foot;  a  surface  45  feet  long  and  1  foot  wide  contains  45 
limes  1  square  foot,  or  45  square  feet;  hence  a  floor  45  feet  long 
and  35  feet  wide  contains  35  times  45  square  feet,  or  1,575 
square  feet. 

Problem:  Measure  this  block  and  determime  how  many  square 
rods  there  are  in  it. 


Solution: 


168  square  feet. 
168 


1344 
1008 
168 

272^)28224  square  feet. 
4  4 


1089)112896(103TVT  square  rods. 
1089 
3996 
3267 

729     243     _81_ 
1089    363  ~  121 

1(58  feet. 


/Vi  square  feet. 


W  THE  SUPPLY. 

Explanation:  Having  nothing  but  a  ten-foot  pole  with  which 
to  measure,  I  determined  the  dimensions  of  the  block  in  feet 
only,  168  feet  square.  A  block  one  foot  square  contains  one 
square  foot;  a  block  168  feet  long  and  one  foot  wide  contains 
168  square  feet;  and  a  block  168  feet  long  and  168  feet  wide 
contains  168  times  168  square  feet,  or  28,224  square  feet.  There 
are  30^  times  9  square  feet,  or  272^  square  feet,  in  a  square  rod; 
in  28,224  square  feet  there  are  as  many  square  rods  as  272^ 
square  feet  are  contained  times  in  28,224  square  feet,  or  103TVT 
times  one  square  rod, /or  103^^-  square  rods  in  the  block. 

Problem:  Measure  this  field  and  cut  off  20  acres  from  the 
west  end. 

Solution:  160    square  rods. 

20 
80)3200  square  rods. 

40  rods. 
40  rd. 


Explanation:  I  measured  the  width  of  the  west  end  with  a  pole 
which  I  prepared  for  the  purpose,  one  rod  in  length,  and  found 
it  to  be  80  rods.  There  are  160  square  rods  in  one  acre,  in  20 
acres  there  are  20  times  160  square  rods,  or  3,200  square  rods. 
The  end  of  the  field  being  80  rods  in  length,  if  a  strip  were  cut 
off  1  rod  wide,  its  area  would  be  80  square  rods,  and  as  the 
strip  is  to  contain  3,200  square  rods,  it  must  be  as  many  times  1 
rod  wide  as  80  square  rods  are  contained  times  in  3,200  square 
rods,  or  40  times  1  rod  wide,  or  40  rods  wide. 

TESTS. 

Add  7  square  miles,  9  acres,  14  square  rods,  5  square  yards, 
8  square  feet,  and  18  square  inches;  10  square  miles,  4  acres, 
27  square  rods,  18  square  yards,  6  square  feet,  and  65  square 
inches;  4  square  miles,  28  acres,  17  square  rods,  20  square  yards, 
4  square  feet,  and  103  square  inches;  1  square  mile,  93  acres, 
18  square  rods,  4  square  yards,  5  square  feet,  and  111  square 
inches. 


TABULATIONS.  J»7 

Multiply  2  square  miles,  40  acres,  18  square  rods,  19  square 
yards,  and  4  square  feet  by  23.  What  is  10  per  cent  of  the 
product? 

I  sold  a  lot  for  $850,  which  was  f  of  what  I  received  for  95 
acres  of  land ;  how  much  was  the  selling  price  of  the  land  per 
acre? 

If  a  field  40  rods  wide  and  100  rods  long  is  headed  by  two 
headers  in  three  days,  how  wide  is  the  field  that  i,s  250  rods  long 
and  is  headed  by  three  headers  in  one  and  one-half  days? 

At  15  cents  per  square  yard,  what  will  it  cost  to  plaster  a 
room  18  feet  long,  15  feet  wide,  and  12  feet  high,  if  an  allowance 
be  made  for  two  doors  7  feet  by  2^  feet,  three  windows  8  feet  by 
2^  feet,  and  a  base  board  one  foot  wide. 

Reduce  235,452  square  inches  to  integral  higher  denominations. 

A  rectangular  farm  is  160  rods  long  and  75  rods  wide;  what 
is  it  worth  at  $65  per  acre?  At  20  per  cent  less  than  $65  per 
acre? 

CUBIC  MEASURE. 

The  table  of  cubes  and  the  idea  of  cubical  measurements  can 
be  developed  most  thoroughly  only  by  having  a  large  quantity 
of  cubical  inches  and  a  foot-rule  with  which  to  measure  them  and 
their  combinations  when  necessary. 

T.     (Giving  each  pupil  a  cubical  inch.)     What  have  you  ? 

C.     I  have  a  cube. 

T.     How  long  is  each  edge? 

C     Each  edge  is  one  inch  long. 

T.     Then  what  kind  of  a  cube  have  you  ? 

C.     I  have  an  inch  cube. 

T.     Yes;    or  we  will  call  it  a  cubic  inch. 

T.  Now,  Mary,  Benjamin,  Reuben,  and  Ida,  (Giving  them 
more  cubes.)  you  may  make  a  square  with  four  cubic  inches. 

C. 


T.  How  long  is  the  square? 

C.  It  is  two  inches  long. 

T.  How  wide? 

C.  Two  inches. 


98 


THE  SUPPLY. 


T.     You  may  now  place  four  more  cubic  inches  upon  these 
four  in  the  same  manner. 
C. 


T. 
C. 
T. 
C. 
T. 
C. 
T. 
C. 
T. 
C. 
T. 


How  many  cubic  inches  have  you  now  in  the  pile? 

1  have  8  cubic  inches  in  the  pile. 
How  long  is  the  pile? 

2  inches. 
How  wide? 
2  inches. 
How  high? 
2  inches. 

How  much  are  2  times  2  times  2? 
2  times  2  times  2  are  8. 

You  may  now  place  four  more  cubic  inches  upon  these 
eight  in  the  same  manner. 
C. 


T.  How  long,  how  wide,  and  how  high  is  the  pile  now? 

C.  It  is  2  inches  long,  2  inches  wide,  and  3  inches  high. '", 

T.  How  many  cubic  inches  does  it  contain  ? 

C.  It  contains  12  cubic  inches. 

T.  2  times  2  times  3  are  how  many  ? 

C.  2  times  2  times  3  are  12. 

T.  You   may  now  take  down  this  pile,   and  Charles,    Ella, 

Edgar,  and  Stella,  may  place  9  cubic  inches  in  the  form  of  a  square. 

C. 


TABULATIONS. 


99 


T.  How  long,  how  wide,  and  how  high  is  the  pile? 

C.  It  is  3  inches  long,  3  inches  wide,  and  1  inch  high. 

T.  How  many  cubic  inches  does  it  contain  ? 

C.  It  contains  9  cubic  inches. 

T.  3  times  3  times  1  are  how  many  ? 

C.  3  times  3  times  1  are  9. 

T.  You  may  now  place  nine  more  upon  these  nine  in  the 
same  manner.  , 

C. 


T. 
C. 

high. 
T. 
C. 
T. 
C. 
T. 


What  are  the  length,  width,,  and  height  of  the  pile  now? 
The  pile  is  now  3  inches  long,  3  inches  wide,  and  2  inches 


How  many  cubic  inches  does  it  contain  ? 
It  contains  18  cubic  inches. 
3  times  3  times  2  are  how  many? 
3  times  3  times  2  are  18. 

A.  pile  3  inches  long,  3  inches  wide,  and  3  inches  high, 
contains  how  many  cubic  inches? 

C.     It  contains  3  times  3  times  3,  or  27,  cubic  inches. 
T.     A  pile  4  inches  long,  4  inches  wide,  and  4  inches  high, 
contains  how  many  cubic  inches? 

C.     It  contains  4  times  4  times  4,  or  64,  cubic  inches. 
T.     A  pile  10  inches  long,  10  inches  wide,  and  10  inches  high, 
contains  how  many  cubic  inches? 

C.     It  contains  10  times  10  times  10,  or  1,000,  cubic  inches. 
T.     A  pile  12  inches  long,  12  inches  wide,  and  12  inches  high, 
contains  how  many  cubic  inches? 

C.     It  contains  12  times  12  times  12,  or  (hesitates.) 
T.     You  may  use  your  slates. 

C.     It  contains  12  times  12  times  12,  or  1,728,  cubic  inches. 
T.     Then  a  pile  one  foot  long,  one  foot  wide,  and  one  foot 
high,  contains  how  many  cubic  inches? 
C.     It  contains  1,728  cubic  inches. 


100  THE  SUPPLY. 

T.     Then  1  cubic  foot  contains  how  many  cubic  inches? 

C.     1  cubic  foot  contains  1,728  cubic  inches. 

T.  We  will  place  this  fact  where  we  can  keep  it  as  part  of 
another  table. 

T.  A  cubical  pile  2  feet  long,  2  feet  wide,  and  2  feet  high, 
contains  how  many  cubic  feet? 

C.     It  contains  2  times  2  times  2,  or  8,  cubic  feet. 

T.  A  cubical  pile  3  feet  long,  3  feet  wide,  and  3  feet  high, 
contains  how  many  cubic  feet  ? 

C.     It  contains  3  times  3  times  3,  or  27,  cubic  feet. 

T.  Then  a  cubical  pile  1  yard  long,  1  yard  wide,  and  1  yard 
high,  contains  how  many  cubic  feet? 

C.     It  contains  27  cubic  feet. 

T.     Then  1  cubic  yard  contains  how  many  cubic  feet? 

C.     1  cubic  yard  contains  27  cubic  feet. 

T.     We  will  place  this  fact  also  as  a  part  of  our  table. 

T.  General  Bidwell  has  some  cord-wood  a  short  distance 
from  here;  after  school  we  will  go  where  it  is,  and  you  may  take 
your  foot-rule  along  with  you  to  measure  some  of  the  piles. 

T.  (In  the  wood-yard  after  school.)  This  is  a  cord  of  wood; 
you  may  now  measure  its  length,  width,  and  height. 

C.     It  is  8  feet  long,  4  feet  wide,  and  4  feet  high. 

T.     How  many  cubic  feet  does  it  contain? 

C.     It  contains  8  times  4  times  4,  or  128,  cubic  feet. 

T.     Then  how  many  cubic  feet  does  1  cord  of  wood  contain  ? 

C.     1  cord  of  wood  contains  128  cubic  feet. 

T.     We  will  now  complete  our  table  with  this  fact: 

CUBIC  MEASURE. 

1728  cubic  inches =1  cubic  foot. 
27  cubic  feet      =1  cubic  yard. 
128  cubic  feet      =1  cord  of  wood. 

T.  You  may  now  measure  these  3  tiers  of  wood,  ascertain 
how  many  cords  there  are  in  each,  and  report  to  the  class 
to-morrow. 

T.  (At  beginning  of  next  day's  recitation.')  James,  you  may 
give  your  report  upon  the  results  of  your  measurements  of  the 
first  tier  of  wood  last  evening. 


TABULATIO\*.  101 

J.      Solution:  28  cubic  feet. 

168 
4 


128)672(5$-  cords. 
640 


Explanation:  I  measured  the  first  tier  and  found  it  to  be  28 
feet  long,  6  feet  high,  and  4  feet  wide.  A  pile  1  foot  long,  1  foot 
wide,  and  1  foot  high,  contains  1  cubic  foot;  a  pile  28  feet  long,  1 
foot  wide,  and  1  foot  high,  contains  28  times  1  cubic  foot,  or  28 
cubic  feet;  a  pile  28  feet  long,  6  feet  high,  and  1  foot  wide,  con- 
tains 6  times  28  cubic  feet,  or  168  cubic  feet;  and  a  pile  28  feet 
long,  6  feet  high,  and  4  feet  wide,  contains  4  times  168  cubic  feet, 
or  672  cubic  feet.  There  are  128  cubic  feet  in  1  cord ;  in  672 
cubic  feet,  there  are  as  many  cords  as  128  cubic  feet  are  contained 
times  in  672  cubic  feet,  or  5^  times  1  cord,  or  5^  cords. 

T.  Gladys,  you  may  place  your  solution  of  the  second  upon 
the  blackboard,  and  with  it  you  may  write  your  explanation,  that 
we  may  criticize  both  the  solution  and  the  explanation. 

T.  Carrie,  you  may  place  your  solution  of  the  third  upon  the 
blackboard  and  explain  it  orally. 

The  following  will  indicate  the  character  of  the  applications 
of,  cubic  measure: 

Problem:  How  long  must  I  make  a  tier  of  4-foof  wood,  which 
is  7  feet  high,  that  it  may  be  worth  $280  at  $8  per  cord? 

Solution:  $8)$280 

35  cords. 
128 


28)4480  cubic  feet  (160  feet. 
28 
168 
168 


Explanation:  Since  the  wood  is  worth  $8  per  cord,  as  many 
cords  are  required  to  be  worth  $280  as  $8  are  contained  times  in 
$280,  or  35  times  1  cord,  or  35  cords.  There  are  128  cubic  teet 


102  THE  SUPPLY. 

in  1  cord;  in  35  cords,  therefore,  there  are  35  times  128  cubic 
feet,  or  4,480  cubic  feet.  A  pile  of  wood  1  foot  Inog,  1  foot  wide, 
and  1  foot  high  contains  1  cubic  foot;  a  pile  of  wood  1  foot  long, 
4  feet  wide,  and  1  foot  high  contains  4  times  1  cubic  foot,  or  4 
cubic  feet;  a  pile  of  wood  1  foot  long,  4  feet  wide,  and  7  feet 
high  contains  7  times  4  cubic  feet,  or  28  cubic  feet;  to  contain 
4,480  cubic  feet,  therefore,  the  pile  must  be  as  many  times  1  foot 
long  as  28  cubic  feet  are  contained  times  in  4,480  cubic  feet,  or 
160  times  1  foot,  or  160  feet,  long. 

TESTS. 

Reduce  25  cords  and  18  cubic  feet  to  cubic  inches. 

A  rectangular  cistern  is  18  feet  deep  and  4  feet  wide;  how  long 
is  it,  if  it  holds  100  barrels  of  water  ? 

If  a  box  is  6  feet  long  and  5  feet  wide,  how  deep  must  it  be  to 
hold  2000  bushels?  To  hold  50  per  cent  more  than  2000  bushels? 

A  vessel  holds  150  barrels  of  water;  it  will  hold  how  many 
bushels  of  wheat? 

A  closed  box  that  is  made  of  inch  lumber  is  4  feet  long,  3  feet 
wide,  and  2  feet  deep,  on  the  inside;  what  is  the  lumber  of  the 
box  worth  at  $2  per  C?  What  is  the  value  of  the  wheat  it  will 
hold,  at  90  cents  per  bushel  ? 

How  much  is  the  lumber  of  an  open  box  worth  that  is  8  feet 
long,  7  feet  wide,  and  4  feet  high,  on  the  outside,  if  the  lumber  be 
2  inches  thick,  and  cost  $14  per  M  ? 

How  long  must  a  stick  of  timber  be,  that  is  18  inches  square, 
to  contain  48  cubic  feet?  To  contain  75  per  cent  of  48  feet? 

What  is  the  value,  at  $2.50  per  C,  of  a  plank  30  feet  long,  3 
inches  thick,  and  14  inches  wide  at  one  end  and  19  inches  wide  at 
the  other  end? 

How  may  you  determine  approximately,  without  weighing,  the 
number  of  tons  of  hay  in  a  mow  or  a  stack? 


PART   II. 


THE   DEMAND. 


"The  educated  man  finds  recreation  in  change  of  work." 

— Harris. 


PERCENTAGE— PROFIT  AND  LOSS.  105 


PERCENTAGE. 


When  percentage  In  the  abstract  is  inseparably  associated 
with,  and  clearly  conceived  to  be,  fractions  in  a  different  form 
only,  and  when  the  pupils  can  think  as  readily  and  express  them- 
selves as  clearly  in  percentage  words  and  symbols,  then  and  not 
till  then  should  the  applications  of  percentage  be  approached. 
These  probably  should  be  taken  up  in  something  of  the  following 
order,  owing  to  the  relative  simplicity  and  importance  of  the 
subjects: 

Profit  and  Loss,  Business  Discount,  Insurance,  Commission, 
Taxes,  Stocks,  Interest,  and  Banking. 

PROFIT  AND  LOSS. 

The  pupils  should  be  asked  to  investigate  the  subject  of  Profit 
and  Loss  for  themselves,  and  to  report  the  names  of  persons 
who  have  made  a  profit  or  a  loss,  the  character  of  the  property 
upon  which  the  profit  or  loss  was  made,  and  the  amount  of  the 
profit  or  the  loss.  From  the  contributions  of  the  pupils,  sufficient 
data  may  be  selected  and  classified  to  enable  the  class  to  thor- 
oughly comprehend  the  subject.  They  will  discover  that  on 
certain  articles  much  higher  rates  of  profit  are  made  than  on 
others;  they  will  discover  that  on  the  same  kinds  of  goods  much 
higher  rates  are  made  on  the  sale  of  small  quantities  than  on  the 
sale  of  large  ones;  and  they  may  perhaps  discover  that  in 
different  localities  the  same  articles  when  sold  in  the  same 
quantities  will  be  sold  at  different  rates  of  profit.  Su9h  dis- 
coveries awaken  an  interest  in  the  subject  that  will  tend  to  an 
exhaustive  research  and  a  finally  firm  hold  of  the  principles  of  the 
subject.  The  language  of  the  problems  will  be  familiar  to  them 
after  this  preliminary  study,  and  the  pupils'  mastery  of  them  will 
be  no  longer  doubtful. 


106  THE  DEMAND. 

To  place  the  pupils  in  ready  command  of  the  material  they 
have  gathered,  and  to  avoid  much  random  work  at  first,  the 
teacher  should  put  a  few  pointed  questions: 

T.  What  do  business  men  compute  their  profit  or  loss  upon 
as  a  base? 

C.     They  compute  their  profit  or  loss  upon  the  cost  as  a  base. 

T.  I  bought  calico  at  4  cents  a  yard,  and  I  wish  to  sell  it  at 
a  profit  of  25  per  cent;  for  how  much  must  I  sell  it  per  yard? 

C.  You  must  sell  it  for  25  per  cent,  or  one-fourth,  more  than 
4  cents  per  yard,  or  you  must  sell  it  for  5  cents  per  yard. 

T.  A  merchant  who  is  making  a  profit  of  20  per  cent  asks  84 
cents  a  gallon  for  sirup;  how  much  did  it  cost  him? 

C.  He  is  selling  it  for  20  per  cent,  or  i  of  the  cost,  more  than 
he  paid  for  it;  therefore  84  cents  is  f  of  the  cost;  one-fifth  of  the 
cost  is  one-sixth  of  84  cents,  or  14  cents;  and  five-fifths  of  the 
cost  is  5  times  14  cents,  or  70  cents. 

T.  Mr.  Burnham  sells  a  suit  for  $22  that  cost  him  $20;  what 
is  his  rate  per  cent  of  profit  ? 

C.  His  profit  is  $2,  or  two-twentieths,  or  one-tenth,  of  the 
cost,  that  is,  ten  per  cent  of  the  cost. 

T.  James  paid  $100  for  a  horse;  he  sold  the  horse  to  John  at 
a  profit  of  10  per  cent;  John  sold  the  horse  to  Oscar  at  a  profit 
of  10  per  cent;  what  did  Oscar  pay? 

C.  John  paid  10  per  cent,  or  one-tenth,  of  $100  more  than 
James;  John  therefore  paid  $110;  Oscar  paid  10  per  cent,  or 
one-tenth,  of  $110  more  than  John  paid,  or  $121. 

TESTS. 

I  bought  5  horses  at  $95  per  head,  and  sold  them  at  a  profit 
of  5  per  cent;  for  how  much  did  I  sell  them  per  head,  and  how 
much  did  I  gain? 

I  bought  85  centals  of  wheat  at  $1.28  per  cental;  I  sold  it  at  a 
profit  of  3  per  cent,  and  with  the  money  bought  a  horse  and  cart, 
paying  twice  as  much  for  the  horse  as  for  the  cart;  how  much 
did  I  pay  for  each  ? 

I  bought  240  acres  and  18  square  rods  of  land  at  $75  per  acre, 
and  sold  the  same  land  for  $9,200;  what  was  my  gain  or  loss  per 
cent? 

Having   bought   a   horse   I   sold    him  again   for  25   per  cent 


PERCENTAGE— BUSINESS  DISCOUNT.  107 

more  than  I  gave  for  him,  and  the  person  to  whom  I  sold  him 
subsequently  disposed  of  him  at  a  loss  of  10  per  cent,  receiving 
$480  for  him;  what  did  the  horse  cost  me,  and  for  how  much  did 
I  sell  him  ? 

An  importer  sold  cloth  to  a  wholesale  dealer  at  an  advance 
of  10  per  cent;  the  wholesale  dealer  sold  it  to  a  retail  dealer  at  an 
advance  of  10  per  cent;  and  the  retail  dealer  sold  it  at  an  advance 
of  20  per  cent.  The  retail  dealer  received  $725;  what  did  it 
cost  the  importer? 

I  bought  a  farm  from  C  at  12  per  cent  more  than  he  paid  for 
it,  and  I  sold  it  to  B  at  5  per  cent  more  than  I  paid  for  it;  B  sold 
it  to  D  at  10  per  cent  less  than  he  paid  for  it,  D  paying  $1,200; 
how  much  did  C  pay? 

"A  merchant  sold  50  bushels  of  wheat  for  A.  H.  Randall  and 
00  bushels  for  William  Earll,  receiving  90  cents  per  bushel  for 
the  lot.  Allowing  that  Mr.  Randall's  wheat  was  worth  20  per 
cent  more  per  bushel  'than  Mr.  Earll's,  how  ought  the  money 
to  be  divided? 

G.  W.  Dorn  invested  a  certain  sum  of  money  in  flour,  and 
Thomas  Bicknell  invested  twice  as  much;  it  transpired  that  Mr. 
Dorn  lost  10  per  cent  and  that  Mr.  Bicknell  gained  10  per  cent, 
and  that  the  difference  between  what  Mr.  Dorn  received  for  his 
lot  and  what  Mr.  Bicknell  received  for  his  was  $260;  how  much 
did  each  invest? 

BUSINESS   DISCOUNT. 

When  the  subject  of  Business  .Discount  is  taken  up,  the  students 
should  again  go  to  the  business  world  to  ascertain  how  and  why 
these  allowances  are  made.  They  will  doubtless  have  gathered 
much  on  this  subject  while  investigating  the  subject  of  profit  and 
loss.  They  will  now  learn  by  whom  and  under  what  circum- 
stances it  is  allowed.  They  will  discover  that  the  time  of  pay- 
ment is  in  reality  the  essential  fact,  though  time  in  a  very 
definite  sense  does  not  enter  into  the  transaction.  They  will 
learn  the  value  and  the' advantage  of  prompt  payment.  They  will 
also  learn  that  sales  in  large  quantities  are  subject  to  greater  pro- 
portionate discount  than  sales  in  small  quantities.  Out  of  all  the 
facts  gathered,  if  they  be  classified  and  discussed,  the  students 
will  form  some  clear-cut  ideas  of  the  laws  of  trade,  the  relations 
of  labor  and  capital,  and  the  principles  of  supply  and  demand. 


108  THE  DEMAND. 

A  few  questions  will  be  found  advantageous  to  place  the  pupil 
on  easy  terms  with  the  subject: 

T.  What  is  the  base  upon  which  business  discount  is 
reckoned  ? 

C.     The  base  is  the  catalogue,  or  marked,  price  of  the  goods. 

T.  State  some  of  the  circumstances  which  lead  to  the  allow- 
ance of  business  discount. 

C.  If  a  store  does  a  credit  business,  a  cash  customer  is  allowed 
a  certain  per  cent  off  for  cash;  a  person  who  has  been  charged 
credit  rates  is  allowed  a  certain  per  cent  off  if  he  pays  sooner 
than  the  terms  of  the  credit  contemplated;  large  houses  publish 
catalogues  for  the  benefit  of  their  customers  who  may  live  at 
remote  points;  a  general  reduction  of  the  prices  therein  published 
may  result  in  their  still  using  the  same  catalogue  with  uniform 
discounts  on  the  prices  therein,  and  then  there  will  be  the 
further  usual  discounts  for  cash,  etc. 

T.  Mr.  Roberts  asks  $40  for  a  suit  df  clothes,  and  allows  5 
per  cent  discount  for  cash;  what  does  the  suit  cost  a  cash 
customer? 

C.  The  suit  costs  $38;  for  5  per  cent,  or  one-twentieth,  of 
$40,  is  $2,  the  discount  which  he  allows. 

T.  Mr.  Brunner  of  Sacramento  has  a  furniture  catalogue 
upon  whose  prices  he  allows  uniformly  a  5  per  cent  discount,  and 
5  per  cent  off  for  cash;  how  much  must  a  cash  customer  pay  for 
a  bill  of  furniture  whose  catalogue  rates  aggregate  $200. 

C.  He  will  pay  $181.50;  for  the  asking  price  is  one-twentieth 
less  than  $200,  or  $190,  and  the  cash  rates  are  one-twentieth 
of  $190,  or  $9.50,  less  yet. 

INSURANCE. 


"One  of  the  chief  characteristics  of  a  good  method  consists  in 
enabling  learners  to  dispense  with  the  assistance  of  a  teacher 
when  they  are  capable  of  self  government." — Marcel. 


The  subject  of  Insurance  is  another  of  the  applications  of  per- 
centage that  requires  much  contact  on  the  part  of  the  pupils 
with  those  engaged  in  it  practically,  before  they  can  clearly  com- 
prehend its  aims  and  purposes.  Polices  of  insurance  of  different 


PERCENTAGE— COMMISSION.  109 

kinds  should  be  careiully  read  and  discussed  in  the  class  and  out 
of  it.  The  common  purpose  of  protection  needs  to  be  firmly  fixed 
by  a  clear  seeing  of  all  the  motives  and  purposes  of  the  parties  to 
the  insurance.  The  greatest  difficulty  usually  encountered  is  to 
establish  the  mutuality  of  the  benefit.  Why  should  a  person 
insure?  Since  it  is  a  fact  that  a  person  pays  a  premium  much  in 
excess  of  the  risk  taken,  why  does  he  insure  at  all?  To  answer 
that  question  properly  is  no  easy  matter;  yet  it  is  a  question  that 
will  as  certainly  arise  as  the  subject  is  properly  investigated. 
The  element  of  protection  will  then  be  found  to  have  a  broader 
signification  and  a  deeper  meaning  than  at  first  we  had  given  it. 
It  will  be  found  that  insurance  is  not  only  a  protection  of  what 
we  have,  but  also  of  what  we  desire  to  have;  not  only  of  present 
conditions,  but  of  future  hopes,  expectations,  and  ambitions  even. 
What  a  field  of  thought  is  opened  up  ?  Can  it  be  doubted  that, 
with  this  budding  wisdom,  the'  students  will  grasp  with  a  firmer 
hand  the  problems  and  processes  that  affect  such  a  subject? 

The  subject  of  insurance  should  be  outlined  somewhat,  so  that 
none  of  the  salient  points  will  be  over-looked  by  the  students  in 
their  tour  of  investigation: 

1.  Ascertain  the  risks  that  are  subjects  of  insurance. 

2.  Ascertain  the  rates  and  times  of  insurance. 

3.  Ascertain  the  ways  in  which  both  parties  are  benefited. 

4.  What  is  the  difference  between  life  insurance  and  endow- 
ment ? 

5.  What  is  the  difference  between  marine  and  fire  insurance? 

6.  Learn  about  insurance  as  regards  tornadoes,  thunder  and 
lightning,  earthquakes,  etc. 

7.  Learn  about  incendiaries. 

8.  Learn  about  the  rates  on  different  kinds  of  buildings. 

9.  Learn  about  the  effect  on  rates,  by  contiguity  of  buildings. 
10.     Learn  about  the  difference  in  rates  upon  buildings  of  the 

same  kind,  located  in  the  city  or  in  the  country. 

COMMISSION. 

v 

Since  it  has  been  found  profitable  to  let  the  pupils  investigate 
the  other  subjects  for  themselves  it  will  not  be  thought  to  be 
other  than  profitable  to  do  likewise  with  Commission.  The 
students  will  ascertain  who  of  their  friends  and  acquaintances 


110  THE   DEMAND. 

ever  received  or  paid  a  commission,  and  upon  what.  A  close 
investigation  will  bring  out  the  fact  that  almost  every  one  has 
received  or  paid  a  commission.  It  will  be  discovered  that  com- 
mission is  a  given  proportion  of  the  amount  for  which  goods 
are  bought  or  sold,  of  the  amount  of  money  collected,  of  the 
amount  handled,  etc.  These  facts  will  lead  to  the  thought  that 
people  are  mutually  dependent  upon,  and  helpful  to,  each  other; 
that  the  measure  of  their  usefulness  and  success  is  their  own 
ability,  industry,  and  opportunity;  and  that  the  opportunity  de- 
pends largely  upon  the  tact  and  temperament  of  the  individual. 

CLASS  WORK. 

Problem:  My  agent  has  purchased  wheat  for  me  to  the  amount 
of  $1728;  what  is  his  commission  at  one  and  one-half  per  cent? 

Solution:  $17.28 

8.64 
$25.92  commission. 

Explanation:  Commission  is  a  certain  per  cent  of  the  amount 
for  which  goods  are  bought  or  sold.  One  per  cent  of  $1728  is 
$17.28,  and  one-half  per  cent  is  one-half  of  $17.28,  or  $8.64; 
hence  one  and  one-half  per  cent  of  $1728,  or  his  commission,  is 
the  sum  of  $17.28  and  $8.64,  or  $25.92. 

Problem:  I  have  sent  to  my  agent  in  Stockton,  California, 
$10,000  to  be  expended  in  the  purchase  of  flour,  after  deducting 
his  commission  of  2  per  cent;  what  will  his  commission  amount 
to,  and  what  will  be  the  amount  of  money  invested  in  flour  ? 

Solution:      1.02)  10000. 00($9803. 92= amount  invested  in  flour. 
918  $196. 08= agent's  commission. 

820 
816 
400 
306 


340 

918 
220 

Explanation:    On   every  dollar  invested  in  flour  the  agent  is 


PERCENT  A  GE—  TA  XKH.  1 1 1 

entitled  to  2  cents  as  commission;  therefore  every  dollar  invested 
in  flour  will  require  $1.02  of  the  $10,000;  hence  there  will  be 
bought  as  many  dollars'  worth, of  flour  as  $1.02  is  contained  times 
in  $10,000,  or  $9,803.92,  the  amount  invested  in  flour.  The 
agent's  commission  is  the  difference  between  $10,000  and 
$9,803.92,  or  $196.08. 

TESTS. 

I  bought  4  head  of  cattle  at  $40  per  head,  and  an  agent  sold 
them' for  me  at  a  profit  of  25  per  cent;  the  agent's  charges  were 
2  per  cent  for  selling;  what  was  the  net  amount  I  received  for 
the  cattle? 

I  sold  through  an  agent  25  horses  at  $150  per  head;  I  directed 
the  agent  to  purchase  wheat  with  the  money  received,  after  de- 
ducting 1  per  cent  for  selling  the  horses  and  2  per  cent  for  the 
purchase  of  the  wheat;  he  purchased  wheat  at  $1.25  per  cental; 
how  many  tons  did  he  buy  for  me? 

My  agent  sent  me  $2,000  as  the  net  proceeds  of  the  sale  of 
wheat  for  me  at  $1.30  per  cental;  his  commission  being  1  per 
cent,  how  many  centals  of  wheat  did  he  sell  for  me? 

I  sell  320  acres  of  land  at  $75  per  acre;  this  price  is  at  a  profit 
of  25  per  cent  upon  the  price  I  paid  for  it.  When  I  bought  the 
land,  I  bought  it  with  the  money  which  I  had  received  as  the  net 
proceeds  of  the  sale  of  barley  by  my  agent  on  a  commission  of  2 
per  cent ;  the  barley  was  sold  at  90  cents  per  bushel ;  how  many 
bushels  of  barley  did  my  agent  sell  for  me? 

Having  85  acres  of  land,  I  placed  it  in  my  agent's  hands,  and 
he  sold  it  for  me  at  $75  per  acre;  he  invested  the  proceeds,  after 
deducting  3  per  cent  commission,  in  cattle  at  $40  per  head,  re- 
taining a  commission  of  2  per  cent  for  the  purchase;  how  many 
cattle  were  purchased? 

TAXES. 

"  Children  like  to  discover  things,  and  to  do  things,  for  them- 
selves, and  they  always  attach  the  highest  value  to  the  knowledge 
which  is  thus  acquired." — Tate. 


The  subject  of  Taxes  will  be,  in  some  respects,  more  interest- 
ing than  any  of  the  subjects  already  investigated.     The  student 


112  THE  DEMAND. 

will  be  surprised,  and  at  first  somewhat  confused,  to  learn  in  how 
many  ways  taxes  are  levied.  He  will  be  surprised  to  learn  that 
all  are  tax-payers,  and  that  no  one  in  all  this  broad  land  can  tell  him 
how  many  dollars  of  taxes  he  pays.  This  remarkable  fact  alone, 
if  properly  discussed  and  considered,  will  impress  the  student  as 
perhaps  no  other  fact  ever  has  before.  He  will  here  and  now 
begin  the  study  of  the  science  of  government.  The  importance 
of  citizenship  will  dawn  upon  him.  The  personal  tax,  the  prop- 
erty tax,  the  internal  revenue  tax,  the  tariff,  can  all  these  be  right 
when  they  differ  so  widely?  In  the  thoughtful  consideration  of 
this  question,  the  student  will  gain  such  a  knowledge  of  taxes  as 
will  preclude  the  possibilty  of  much  doubt  when  applications  are 
presented  for  his  solution.  What  a  fertile  field  arithmetic  has 
become?  What  new  crop  will  it  not  produce?  is  asked. 

The  subject  'of  Taxes  is  so  broad  and  so  complex  that  its  con- 
sideration must  be  given  more  time  and  attention  than  other 
applications  of  percentage.  It  is  not  so  easy  for  the  pupil  to 
collect  data  on  the  subject  as  on  many  others,  owing  to  the  lack 
of  definite  knowledge  by  business  men  themselves;  hence  much 
more  skillful  direction  on  the  part  of  the  teacher  will  be  neces- 
sary. It  will  often  be  found  necessary  for  the  pupils  to  write 
letters  to  friends  at  a  distance,  and  to  resort  to  other  unusual, 
though  very  effective,  methods  of  investigation.  These  extraordi- 
nary efforts  however  will  be  the  very  life  of  the  subject,  and  will 
produce  a  lasting  knowledge  of  the  subject. 

SUGGESTIONS. 

1.  What  methods  are  employed  by  the  United  States  Gov- 
ernment to  raise  money  for  its  support? 

2.  What  methods  are  employed  by  the  State  Government  to 
raise  funds  for  its  support?     By  the  County?     By  the  City? 

3.  How  are  schools  supported  ? 

4.  What  is  the  difference  between    Internal   Revenue  and 
Duties?     How  and  by  whom  are  each  collected? 

5.  What  becomes  of  the  money  derived  from  the  sale  of 
postage  stamps  and  other  postal  matter? 

6.  What  becomes  of  the  money   derived   from   the  sale  of 
public  lands? 

7.  Mention  several  ways  in  which  a  person  helps  to  support 
the  United  States  Government  financially? 


PERCENTAGE— STOCKS.  113 

8.  Upon  what  kinds  of  property  are  State  taxes  paid?     The 
usual  rate? 

9.  When  are  assessments  made?     By  whom? 

10.  When  are  taxes  collected?     By  whom? 

11.  How  about  poll  taxes?     Road  taxes? 

12.  City  taxes?     Street  improvements? 

STOCKS. 


"All  learning  is  a  translation  of  an  unknown  into  a  known." 

— Harris. 


The  next  subject  that  naturally  suggests  itself,  as  being  next  in 
importance  and  interest,  as  well  as  in  the  natural  gradation  in 
point  of  difficulty,  is  that  much  abused,  because  badly  taught, 
subject  of  Stocks.  This  subject  cannot  be  considered  by  the 
pupils  without  at  the  same  time  considering  the  subject  of( 
Partnership,  so  closely  are  they  allied.  Careful  investigation 
will  establish  such  points  of  difference  as  the  following:  Stock 
Companies  have  officers  who  transact  the  business;  Partnerships 
have  not.  Stock  Companies  continue  for  a  limited  time;  Partner- 
ships are  formed  for  an  unlimited  time.  In  Stock  Companies  the 
shareholders  are  liable  for  the  debts,  in  proportion  to  the  amount 
of  their  stock ;  in  Partnership  the  partners  are  each  liable  for  all 
the  debts.  These  points  being  established,  the  students  will 
naturally  segregate  the  business  concerns  with  which  they  are 
familiar,  into  Stock  Companies  and  otherwise.  They  should  then 
be  sent  to  learn  all  they  can  about  Stock  Companies,  their 
organization,  their  papers,  their  manner  of  doing  business,  their 
disposition  of  profits,  the  purchase  and  sale  of  proprietary  inter- 
ests, etc.  In  the  course  of  this  investigation,  they  will  discover 
that  the  share  is  the  unit  of  all  the  business  calculations,  and  that 
knowing  the  entire  number  of  shares  and  the  number  owned  by 
any  individual,  his  interest  in  the  property  and  the  profits,  or  his 
liability,  in  the  contingency  of  a  loss,  is  easily  ascertained.  They 
will  also  discover  that  the  true  value  of  a  share  bears  little  if  any 
relationship  to  the  par  value  of  a  share,  that  the  par  value  of  a 
share  is  generally  $100,  and,  when  otherwise,  that  it  is  an  aliquot 
part  or  a  multiple  of  $100.  They  will  also  discover,  with  slight 


114  .  THE  DEMAND. 

aid,  that  all  quotations,  dividends,  assessments,  and  charges,  are 
(when  the  par  value  is  $100)  as  many  dollars  per  share  as  there  are 
units  in  the  quoted  per  cent;  that  is,  stock  quoted  at  96  is  worth 
$96  per  share;  a  dividend  of  7  per  cent  is  a  dividend  of  $7  per 
share;  if  a  person  charges  one  and  one-half  per  cent  brokerage, 
he  charges  one  and  one- half  dollars  per  share.  Likewise,  if  the 
par  value  be  an  aliquot  part  or  a  muliple  of  $100,  the  number  of 
dollars  per  share  wilt  be  the  same  aliquot  part  or  multiple  of  the 
number  expressing  the  quotation,  assessments,  etc.  Thus  an  8  per 
per  cent  assessment  is  an  assessment  of  $16  per  share  on  stock 
whose  par  value  is  $200,  and  an  assessment  of  $4  per  share  on  stock 
whose  par  value  is  $50  per  share.  It  will  readily  be  seen  that  the 
subject  of  stocks  will  present  no  serious  difficulties  if  the  share  be 
considered  the  unit,  and  if  all  computations  be  on  the  basis 
of  so  many  dollars  per  share  instead  of  so  many  cents  on  the 
dollar.  The  share  is  tangible  but  the  so-called  dollar  of  par 
value  is  not. 

CLASS  WORK. 

Problem:  Having  15  shares  of  S.  P.  R.  R.  stock,  I  ordered 
my  agent  to  sell  it  for  me;  he  sold  the  stock  at  85  and  charged 
me  2  per  cent  brokerage;  how  much  did  I  realize? 

Solution : 


Explanation:  A  broker's  charge  is  a  certain  per  cent  of  the 
par  value  of  stock.  The  broker's  charges  in  this  instance  are 
therefore  $2  per  share.  If  he  sells  the  stock  at  $85  per  share  and 
charges  me  $2  per  share  for  selling  it,  I  shall  realize  $83  per 
share,  and  on  15  shares  I  shall  realize  15  times  $83,  or  $1245. 

Problem:  Wishing  to  purchase  some  stock  of  the  Stockton 
Savings  Bank,  I  sent  my  agent  $920  with  which  to  purchase 
stock  and  pay  his  brokerage.  He  charges  one  per  cent  for  buy- 
ing and  pays  91  for  the  stock;  how  many  shares  does  he  buy 
for  me? 

Solution :  $92)$920 

10  shares. 

Explanation:    Since  the  stock  is  purchased  at  $91  per  share 


PERCENT  A  GE— INTEREST.  1 15 

and  the  broker  charges  $1  per  share  for  his  services,  each  share 
costs  gross  $92,  and  for  $920  he  will  buy  for  me  as  many  shares 
as  $92  are  contained  times  in  $920,  or  ten  times  one  share, 
or  ten  shares. 

Problem.  The  broker's  charges  at  one  and  one-half  per  cent 
amounted  to  $90,  for  the  purchase  of  stock  at  76;  what  was  the 
value  of  the  stock  purchased  ? 

Solution:  $1.50)$90. 00(60  shares.  $76 

900  60 


0  $4560 

Explanation:  As  the  broker's  charges  are  one  and  one-half 
dollars  per  share,  or  $90  in  the  aggregate,  he  must  have  pur- 
chased as  many  shares  as  one  and  one-half  dollars  are  con- 
tained times  in  $90,  or  60  times  one  share,  or  60  shares.  And 
as  the  shares  each  cost  $76,  60  shares  cost  60  times  $76,  or  $4560. 

INTEREST. 

The  subjects  so  far  considered  will  not  have  impressed  the 
student  as  being  dependent  to  any  definite  degree  upon  the  idea 
of  time,  though  in  many  of  them  time  is  an  essential,  though  in- 
definite, element.  In  Interest  however,  time  will  stand  out  as  a 
limiting  element  in  all  transactions.  The  origin  and  history  of 
interest  will  open  an  interesting  field  for  the  awakening  of  desire 
and  the  enlisting  of  effort.  Allusions  to  it  in  the  Bible  and  in 
Ancient  history,  the  church  in  its  relationship  to  it,  its  connection 
with  the  Jewish  persecutions, — all  these  are  prolific  topics  for 
discussion  in  the  class  room.  Different  rates  of  interest  in  differ- 
ent countries  and  in  different  parts  of  the  same  country,  and  the 
causes  therefor,  is  another  question  in  political  economy  whose 
partial  solution  is  within  the  grasp  of  students  at  quite  an  early 
age.  How  their  horizen  will  broaden,  only  such  teachers  as  have 
taught  thus  rationally  are  able  to  tell.  True  and  bank  discount 
are  only  different  methods  of  computing  interest  when  negotiable 
paper  is  being  transferred.  Compound  and  exact  interes^  are 
likewise  only  different  methods  which  are  allowed  for  computing 
interest  in  certain  places  and  under  certain  circumstances;  all 
therefore  should  be  treated  as  varying  phases  of  Interest. 


116  THE  DEMAND. 

CLASS  WORK. 

Problem:    Find  the  interest  of  $725.40  for  2  years,  5  months, 
and  25  days,  at  6  per  cent  per  annum. 


Solution:  .12  $725.40 


1209 

65286 
29016 

7254 
$108.2055 

Explanation:  The  interest  of  one  dollar  for  2  years,  at  6  per 
cent,  is  12  cents,  for  5  months  is  two  and  a  half  cents,  and  for  25 
days  is  four  and  one-sixth  mills;  hence  for  two  years,  five 
months,  and  twenty-five  days,  the  interest  is  14  cents  and  nine 
and  one-sixth  mills..  The  interest  of  $725.40  therefore  is  725.4 
times  $.149i,  or  $108. 21. 

Problem:  What  is  the  amount  of  $235  on  interest  for  1  year, 
9  months,  and  8  days,  at  7  per  cent  per  annum  ? 

Solution : 


$24.988-| 

4.164 
$264. 153  =  Amount. 

Explanation:  The  interest  of  $1  for  one  year  at  6  per  cent  is  6 
cents,  for  9  months  is  four  and  a  half  cents,  and  for  8  days  is  one 
and  one-third  mills;  hence  the  interest  of  $1  for  1  year,  9  months, 
and  8»days  is  $.106^.  The  interest  of  $235  at  6  per  cent  is  235 
times  $.106-3-,  or  $24.988-^;  and  at  7  per  cent,  which  is  one-sixth 
more  than  six  per  cent,  the  interest  is  one-sixth  more,  or  $4. 16 
more;  the  amount,  therefore,  which  is  the  sum  of  the  interest  and 
principal,  is  $264. 15. 

Problem:  Nothing  having  been  paid,  what  is  the  amount  of 
$600  for  2  years  and  3  months  at  9  per  cent  per  annum,  interest 


PERCENTAGE— INTEREST.  117 

payable  annually,  and  if  not  paid  as  it  becomes  due,  the  interest 
to  be  added  to  the  principal,  become  a  part  thereof,  and  bear 
interest  at  the  same  rate? 

Solution:  $1.09 

600 
$654. 
1.09 


$58.86 
654 
$712.86 

1.02J 
178215 
142572 
71286 
$728.89935 

Explanation:  Nothing  having  been  paid,  the  principal  each 
succeeding  year  is  the  amount  of  the  preceding  year.  The 
amount  of  $1  for  one  year  at  9  per  cent  is  $1.09,  and  of  $600  is 
600  times  $1.09,  or  $654.  Likewise  the  amount  of  $654  for  one 
year  at  9  per  cent  is  $712.86.  The  interest  of  $1  for  3  months  at 
9  per  cent  is  two  and  one-fourth  cents;  the  amount  of  $1,  there- 
fore, for  the  same  time  and  at  the  same  rate  is  $1.02^,  and  of 
$712.86  is  712.86  times  $1.02^,  or  $728.90. 

Problem:  In  what  time  will  $560  bear  $70  interest,  at  8  per 
cent  per  annum? 

Solution :     $560 

.08 

$44.8)70.0(lT9e  years  =  1  year,  6  months,  and  22|  days. 
448 

252      63       9 
448  =  112=16 

Explanation:  The  interest  of  $560  for  one  year  at  8  per  cent  is 
$44.80.  $560,  therefore,  will  have  to  bear  interest  as  many  times 
one  year  as  $44.80  is  contained  times  in  $70,  or  one  and  nine- 
sixteenths  times  one  year,  which  equals  one  and  nine-sixteenths 
years,  or  one  year,  six  months,  and  twenty-two  and  one-half 
days. 


118  THE  DEMAND. 

Problem:    The  interest  of  $650   for  1  year   and  6  months  is 
$97.50;    what  is  the  rate  of  interest  it  bears? 

Solution : 


$9. 75)$97.50(10  per  cent. 
975 
0 

Explanation:  The  interest  of  $650  for  one  year  at  one  per 
cent  is  $6.50,  for  6  months  is  $3.25,  hence  for  1  year  and  6 
months  it  is  $9. 75.  Since  the  interest  at  one  per  cent  for  the 
given  time  is  $9.75,  and  since  the  entire  interest  is  $97.50,  the 
rate  must  be  as  many  times  one  per  cent  as  $9.75  is  contained 
times  in  $97.50,  or  10  times  one  per  cent,  or  10  per  cent. 

Problem : 
$825.  Chico,  Cal.,  March  1,  1889. 

One  year  from  date,  for  value  received,  I  promise  to  pay  to 
A.  D. ,  or  order,  Eight  Hundred  Twenty-Five  Dollars,  with 
interest  at  eight  per  cent  per  annum.  C.  D. 


<fc  ^  <J?    ^  O 

«  rT   &  ^     <^  i 


1  - 

5>  \> 

§  ^ 

fc  s 

% 


Os  OQ    Oo~ 

°o    s    s 


C.  D.  having  sent  word  that  he  will  pay  the  note  on  Decem- 
ber 1,  1892,  how  much  will  be  due  and  unpaid  at  that  time? 

Solution:  $825  1  year. 

.08 
66.00 
825. 


$891.       1  year,  4  months,  2  days. 


PERCENT  A  GE— INTEREST. 


$791         2  years,  2  months,  4  days. 
.147^       2  years,  5  months,  17  days. 


.12 

.025 


659-J 
5537 
3164 
791 


.147 


38.978 
$155.91 

791. 
$946.91 

355. 
$591.91 
.0171 
9865 
414337 
59191 
$10.16112 
3.38704 


12 


1 

18 


3  months      1.3  days. 


J.oo 
591.91 


Explanation:  This  note  being  of  the  usual  form  comes  within 
the  purview  of  the  United  States  rule.  The  interest  from  date 
to  the  time  of  the  first  payment  is  $66.  Since  $100  is  the  pay- 
ment, the  principal  will  be  reduced  by  such  an  amount  as  is  not 
required  to  pay  the  interest.  The  sum  of  the  original  principal 
and  the  accrued  interest,  diminished  by  the  payment,  produces 
the  true  amount  of  the  debt  that  remains  unpaid,  or  $791.  The 
time  intervening  to  the  next  payment  is  1  year,  4  months  and  2 
days.  It  is  evident,  by  inspection,  that  at  8  per  cent  the  interest 
will  be  more  than  10  per  cent  of  the  principal,  and,  therefore, 
more  than  the  payment;  as  the  principal  must  not  be  increased 
,  and  as  the  entire  payment,  and  more,  is  required  to  pay  the 
accrued  interest,  it  is  necessary  to  compute  the  interest  until  such 
time  as  the  payments  are  sufficient  to  more  than  pay  the  accrued 
interest. 

Since  the  next  interval  varies  but  slightly  from  the  one  just 


1-20  THE  DEMAND. 

discussed,  and  since  the  payment  is  less  than  half  as  great,  it  is 
evident  that  to  compute  to  the  time  of  the  third  payment  would 
also  be  useless.  The  fourth  payment  is  of  sufficient  magnitude 
to  dispel  all  doubts,  and  to  make  it  certain  that  the  aggregate  of 
the  second,  third,  and  fourth  payments  will  much  more  than  .pay 
the  interest  from  the  time  of  the  first  payment  to  the  time  of  the 
fourth  payment.  The  interest  from  the  time  of  the  first  payment 
to  the  time  of  the  fourth  payment  is  $155.91;  this  added  to  the 
$791,  that  remained  unpaid,  produces  $946.91.  The  sum  of  the 
last  three  payments,  $355,  subtracted  from  $946.91,  leaves 
$591.91.  This  last  amount  is  the  sum  that  will  draw  interest  for 
the  remaining  time,  3  months  and  13  days,  to  the  time  of  settle- 
ment. The  interest  for  3  months  and  13  days  is  $13.55;  the  sum 
C.  D.  must  pay,  therefore,  on  December  1st,  1892,  is  $605.46. 

A  TEST. 

Southworth  &  Grattan  bought  of  Hedges  &  Buck  goods  to 
the  amount  of  $1,000,  payable  in  6  months,  without  interest. 
One  month  afterward,  they  sold  the  goods  for  cash  at  an  advance 
of  10  per  cent,  and  immediately  put  the  money  at  interest  at  6 
per  cent.  When  the  6  months  had  expired,  they  collected  the 
amount  of  the  money  they  had  lent,  and  paid  the  bill  due  Hedges 
&  Buck;  how  much  did  they  gain? 

BANKING. 

"I  am  convinced  that  the  method  of  teaching  which  ap- 
proaches most  nearly  to  the  method  of  investigation  is  incom- 
parably j  the  best. ' '  — Biirke. 


Under  the  comprehensive  term  of  Banking,  the  entire  business 
of  banks  should  be  considered,  so  that  separate  investigations  of 
the  subjects  of  foreign  and  domestic  exchange  will  be  unneces- 
sary. The  subject  of  Banking  will  require  much  skillful  direc- 
tion, great  patience,  ample  time,  and  good  judgment,  in  order 
that  the  results  shall  be  clearly  satisfactory.  How  the  vast 
business  of  the  world  is  transacted  with  only  an  infinitesimal 
portion  of  its  value  in  money  ever  moving  at  all;  who  keep  the 
accounts  that  enable  all  the  settlements  to  be  made  at  stated 


PERCENTAGE— BANKING.  121 

periods  with  such  wonderful  precision;  what  becomes  of  the 
coin  that  is  shipped  from  one  nation  to  another  as  the  balance  of 
trade;  how  coin  that  is  steadily  pouring-  in  certain  directions  is 
replaced:  these  are  living  questions,  the  solution  of  which  raises 
the  student  from  a  lower  to  a  higher  plane,  and  kindles  within 
him  new  desires  and  aspirations.  He  sees  the  wonders  of 
civilization  and  the  possibilities  of  the  human  mind,  and  is  lead  to 
look  with  awe  and  reverence  to  the  power  that  has  fashioned  it. 
In  Banking,  questions  of  the  following  nature  will  lead  the 
pupil  to  investigate  with  more  skill,  having  definite  points  around 
which  to  associate  his  data: 
Bank  Notes: 

How  are  they  written  ? 

How  many  signatures  are  usually  required? 

What  is  the  usual  rate  of  interest? 

At  maturity,  what  action  is  taken  ? 

What  is  the  form  of  a  protest? 
Bank  Discount: 

What  is  the  nature  of  the  notes  which  banks  will  purchase? 

If  they   do   not   bear   interest,  upon   what   is  the  discount 
computed? 

If  they  bear  interest,  upon  what  is  the  discount  computed  ? 
Deposits: 

What  are  "call"  deposits? 

What  kinds  of  deposits  draw  interest? 

How  does  the  rate  of  interest  on  deposits  compare  with  the 
legal  rate? 

What  is  the  difference  between   National   Banks,    ordinary 
Commercial    Banks,    and   Savings    Banks,    as   regards 
deposits  ? 
Drafts  and  Bills  of  Exchange: 

What  is  the  difference  between  Drafts  and  Bills  of  Exchange  ? 

For  what  purpose  are  they  used? 

What  do  we  mean  when  we  say    ' '  sight   drafts ' '  ?     Time 
drafts? 

What  are  the  conditions  of  trade  that  cause  drafts  or  bills 
of  exchange  to  be  at  a  premium  ?    At  a  discount  ? 

Why  are  two  or  more  copies  of  Bills  of  Exchange  given  ? 

Do   Express   and   Telegraph   Companies   do   any  of   these 
kinds  of  business? 


122  THE  DEMAND. 

CLASS  WORK. 

Problem :  At  6  per  cent  per  year,  what  is  the  difference  be- 
tween the  bank  discount  and  the  true  discount  of  a  note  for 
$2,059.40,  payable  in  60  days,  without  interest? 

Solution : 

$20.594  Bank  Discount.  $2059.40 

$20.39  1. 01)2059. 40($2039. 01  pres.  value. 

.20  difference.  202  $20.39  true  dis. 

394 
303 


910 
909 


100 

Explanation:  Banks,  for  the  purposes  of  discount,  reckon  inter- 
est on  the  value  of  the  note  at  maturity.  Therefore,  the  Bank 
Discount  is  one  per  cent  of  $2059.40,  or  $20.594.  True  Discount 
is  the  difference  between  the  present  value  of  a  note  and  its 
value  at  maturity,  and  is  based  on  the  ordinary  method  of  com- 
puting interest.  The  present  value,  therefore,  is  such  a  sum  as 
being  placed  at  interest  at  the  given  rate  and  for  the  given  time 
will  amount  to  the  value  of  the  note  at  maturity.  Since  one 
dollar  in  60  days  at  6  per  cent  will  amount  to  $1.01,  and  since 
$2059.40  is  the  amount  of  the  present  value  for  the  same  time 
and  at  the  same  rate,  the  present  value  must  be  as  many  times 
$1  as  $1.01  is  contained  times  in  $2059.40,  or  $2039.01;  and 
the  true  discount  is  the  difference  between  $2059.40  and  $2039.01, 
or  $20.39;  and  the  difference  between  the  Bank  Discount  and 
the  True  Discount  is  the  difference  between  $20.59  and  $20.39, 
or  $.20,  approximately.  This  difference  is  the  interest  on  the 
True  Discount  for  the  given  time  and  at  the  given  rate. 

TESTS. 

Sidney  Newell  bought  18  shares  of  bank  stock  at  a  premium 
of  8  per  cent  on  their  par  value  of  $100  per  share.  Six  months 
afterward,  and  at  the  end  of  every  subsequent  six  months,  he 
received  a  dividend  of  four  and  one-half  per  cent.  At  the  end 
of  two  years  and  three  months,  he  sold  the  stock  at  a  premium 


PERCENTAGE— BANKING.  123 

of  12  per  cent.  Money  being  worth  8  per  cent  per  year,  com- 
pound interest,  how  much  did  he  gain? 

Having  75  shares  of  bank  stock,  I  sell  it  through  a  broker  at 
76,  paying  1%  brokerage.  With  the  net  proceeds,  I  buy  a  draft 
at  90  days,  on  Denver.  If  sight  drafts  on  Denver  cost  101,  and 
if  interest  is  7%,  what  is  the  face  of  the  draft? 

I  paid  $2400  for  a  60-day  draft;  exchange  being  at  a  premium 
of  l-|-%,  and  the  rate  of  interest  being  8%,  what  was  the  face 
of  the  draft? 

A  pound  sterling  being  worth  $4.8665  in  U.  S.  Gold  Coin, 
how  many  dollars  will  purchase  a  set  of  exchange  to  liquidate  a 
debt  of  ;£200  in  Liverpool,  exchange  selling  in  San  Francisco  at 
3%  premium? 


124  THE  DEMAND. 


LONGITUDE  AND  TIME. 


"Our  age  inclines  at  present  to  the  superstition  that  man  is 
able,  by  means  of  simple  sense- perception,  to  attain  a  knowl- 
edge of  the  essence  of  things,  and  thereby  dispense  with  the 
trouble  of  thinking." — Rosencranz. 


The  object  of  the  study  of  this  subject  is  to  gain  a  clear  con- 
ception of  the  following  facts:  (1)  That  there  are  two  days  of 
the  week  in  existence  at  the  same  time,  and  only  two;  (2) 
That  the  boundary  lines  of  these  days  is  the  one-hundred- 
eightieth  meridian,*  and  the  meridian  containing  the  midnight 
line;  (3)  That  the  lune  that  has  the  one  day  is  constantly 
diminishing  in  width,  until  its  width  is  nothing,  while  the  lune 
containing  the  succeeding  day  of  the  week  is '  constantly  increas- 
ing in  width,  until  its  width  is  co-extensive  with  the  circumference 
of  the  earth;  (4)  That,  at  that  instant,  there  is  but  one  day  of  the 
week  throughout  the  surface  of  the  earth;  (5)  That  the  earth 
revolves  from  the  west  toward  the  east;  (6)  That  west  and 
east  are  only  relative  terms,  not  absolute  directions;  (7)  That, 
therefore,  when  a  person  looks  northward,  the  surface  of  the  earth 
moves  toward  his  right,  but,  if  he  look  southward,  it  moves 
toward  his  left;  (8)  That  a  day  is  that  period  of  time  that 
elapses  from  the  time  a  given  meridian  is  vertically  shone  upon 
by  the  sun  to  the  time  when  the  same  meridian  is  vertically  shone 
upon  the  next  time;  (9)  That  an  hour  is  one  of  the  24  equal  parts 
of  a  day ;  (10)  That  every  circle  is  divided  into  360  equal  parts 
called  degrees;  (11)  That,  therefore,  degrees  vary  in  length  as 
the  circumference  of  the  circle  varies  in  length. 

These  facts  being  definitely  established,  the  applications  of  the 

This  meridian  is  the  "  International  Day-line." 


LONGITUDE  AND  TIME. 


125 


principles  will  not  carry  with  them  the  meaningless  jargon  of 
words  that  tend  to  falsify  in  the  minds  of  students  the  true 
idea  of  education:  that  the  truth  fully  revealed  is  the  end  in 
view. 

As  valuable  aids  to  the  proper  conception  of  this  subject,  it 
will  be  found  that  the  following  or  kindred  diagrams  will  be 
exceedingly  exhaustive,  after  the  general  ideas  have  been 
developed  with  a  globe: 


90-W 


'«0°E.or\V 
Midnight- 


90°  F. 


Noon. 


126 


THE  DEMAND. 


Noon. 


Midnight. 


Noon 
VSO°  F.  or  \V. 


Greenwich- 
Midnight- 


LONGITUDE  AND  TIME. 


127 


Noon. 


Noon. 


*frdnight- 


128  THE  DEMAND. 


oo*  wL 

90«EL 


'**  E. 

Midnight. 


The  figures  should  all  be  drawn  on  a  blackboard  on  the  north 
wall  of  the  school-room,  so  that  the  students  in  looking  at  them 
will  have,  as  nearly  as  may  be,  a  truthful  view  of  the  relative 
directions  of  east  and  west,  and  of  the  relative  locations  of  the 
different  portions  of  the  earth.  These  seven  figures  are  really 
the  same  figure,  representing  different  divisions  of  time.  They 
are  each  a  section  of  the  earth  whose  boundary  is  a  parallel  of 
latitude,  as  all  parallels  are  alike  as  regards  time  of  day,  expressed 
in  hours  and  minutes;  when  conclusions  are  drawn  with  regard 
to  places  on  one  parallel,  they  will  be  conclusions  regarding  the 
entire  surface  of  the  earth. 

In  figure  1,  the  earth  is  represented  as  being  in  such  a  position 
that  the  sun  shines  directly  upon  the  meridian  which  passes 
through  Greenwich.  That  is  the  figure  with  which  to  open  the 
subject,  and  the  one  to  dwell  upon  until  by  dextrous  questioning 
the  following  facts  are  drawn  out: 

(1)  That  east,  to  two  different  persons  living  on  the  opposite 
sides  of  the  earth,  is  directly  opposite  directions,  and  that  east 
at  the  same  place  is  at  no  two  consecutive  minutes  the  same 
absolute  direction;  (2)  That  a  place,  179  degrees,  59  minutes, 
and  59  seconds  east,  has  nearly  midnight  of  Wednesday  afternoon, 
while  one,  179  degrees,  59  minutes,  and  59  seconds  west,  only  a 
few  rods  distant,  has  only  the  very  beginning  of  Wednesday;  in 
other  words,  the  difference  in  time  between  two  such  places,  only 


LONGITUDE  AND  TIME.  129 

a  few  rods  apart,  is  but  an  instant  less  than  24  hours.  Let  these 
two  thoughts  be  dwelt  upon  until  no  doubt  remains.  Then,  by 
skillful  questioning,  establish  the  fact  that  while  this  later  time 
is  in  east  longitude  the  place  that  possesses  it  is  west  of  the  one- 
hundred-eightieth  meridian.  This  is  a  vital  point;  for  it  is 
necessary  that  all  should  fully  comprehend  this  fact  in  order  to 
understand  the  change  in  the  calendar  that  is  made  by  ships  in 
crossing  and  recrossing  the  one-hundred-eightieth  meridian. 

That  fact  being  clearly  established,  figure  2  is  ready  to  be 
studied  as  representing  the  position  of  the  earth  at  a  time  sub- 
sequent to  that  represented  by  figure  1.  Again,  by  skillful  ques- 
tioning, lead  the  pupil  to  see  that  if  the  place  on  the  west  side  of 
the  one-hundred-eightieth  meridian  had  nearly  midnight,  Wednes- 
day, then,  when  the  earth  had  revolved  a  little  further,  the  same 
place  must  necessarily  have  Thursday;  hence,  that  the  new  day 
always  begins  on  the  west  side  of  the  one-hundred-eightieth 
meridian;  and  that  places  on  opposite  sides  of  this  line  as  the 
earth  continues  to  revolve,  must  continue  to  have  nearly  24 
hours  difference  in  time.  , 

Then,  by  a  comprehensive  examination  of  figures  3,  4,  5,  6, 
and  7,  the  further  fact  can  be  established  in  the  minds  of  the 
students,  that  Wednesday  must  wane  as  Thursday  waxes,  that 
is,  that  the  portions  of  the  earth  that  have  Wednesday  must  grow 
less  and  less,  while  those  that  have  Thursday  grow  more  and 
more,  until  when  one  revolution  of  the  earth  shall  have  been 
completed,  Wednesday  will  have  expired  on  the  east  side  of  the 
one-hundred-eightieth  meridian,  and  Thursday  will  be  the  day  for 
all  portions  of  the  earth;  then,  Friday  will  be  born  on  the  west 
side  of  the  one-hundred-eightieth  meridian. 

These  principles  are  fundamental,  and  indispensable  to  a  clear 
conception  of  this  very  interesting  and  useful  subject. 


CLASS  WORK. 


Problem :  The  Longitude  of  A  is  125  degrees  west,  and  of  B  is 
87  degrees  and  30  minutes  east;  what  is  the  time  at  B  when  it  is 
2  o'clock  and  20  minutes  p.  .M.  at  A? 


130 


THE  DEMAND. 


Solution : 


125C 

87C 


30' 


15)212°  3(X 

14  hr.  10  min. 

2  hr.  30  min. 
16  hr.  40  min. 

4  hr.  40  min.   A.  M.,  next  day 


Noon. 


87°  30'  E. 


A  2  hr.  30  min.  P.M. 
125°  W. 


Greenwich. 


Explanation:  Since  A  is  125  degrees  west  of  the  meridian 
which  passes  through  Greenwich,  and  since  B  is  87  degrees  and 
30  minutes  east  of  the  same  meridian,  it  is  evident  that  B  has 
revolved  through  the  sum  of  125  degrees  and  87  degrees  and  30 
minutes,  or  212  degrees  and  30  minutes,  since  it  had  the  same 
time  that  A  now  has.  The  earth  revolves  15  degrees  in  one 
hour;  to  revolve  210  degrees  will  require  14  hours;  to  revolve  2 
degrees  and  30  minutes,  or  150  minutes,  will  require  10  minutes 
of  time,  since  the  earth  revolves  15  minutes  of  longitude  in  one 
minute  of  time.  Therefore,  it  is  14  hours  and  10  minutes  since  B 
had  the  same  time  that  A  now  has;  hence  the  time  that  B  has  is 
16  hours  and  40  minutes  past  noon,  or  4  hours  and  40  minutes 
A.  M.,  the  next  day. 

Problem:  It  is  8  hours  and  40  minutes  A.  M.  at  A  when  it  is  1 
hour  and  20  minutes  A.  M.  at  B,  whose  longitude  is  25  degrees 
and  30  minutes  east;  what  is  the  longitude  of  A? 


LONGITUDE  AND  TIME. 


131 


Solution: 


8  hours  40  minutes. 

1  hour    20  minutes. 

7  hours  20  minutes. 

15 

110° 

25°  30* 

135°  30'  east. 

Noon. 


«S  hr.  40  min.  A.  M. 
(?)  A 


L  hr.  20  min.  A.  M. 
25°  3(X  E. 


Greenwich. 


Explanation:  Since  the  time  at  A  is  8  hours  and  40  minutes, 
and  of  B  1  hour  and  30  minutes,  both  A.  M.  ,  of  the  same  day,  A 
must  have  had  the  same  time  that  B  now  has  7  hours  and  20 
minutes  before;  and,  since  the  earth  revolves  to  the  eastward,  A 
must  lie  east  of  B.  The  earth  revolves  15  minutes  of  longitude 
in  one  minute  of  time;  in  20  minutes,  therefore,  it  revolves  20 
times  15  minutes  of  longitude,  or  300  minutes  of  longitude,  or  5 
degrees.  It  revolves  15  degrees  in  1  hour;  in  7  hours,  it 
revolves  7  times  15  degrees,  or  105  degrees;  hence,  in  7  hours 
and  30  minutes,  it  revolves  105  degrees  plus  5  degrees,  or  110 
degrees;  hence  A  lies  110  degrees  east  of  B,  or  135  degrees 
and  30  minutes  east  of  the  meridian  which  passes  through 
Greenwich. 

TESTS. 

What  are  the  boundary  lines  between  the  days? 
Where  does  the  day  begin  ? 
Where  does  the  day  end  ? 


132  THE  DEMAND. 

What  is  the  position  of  the  earth  with  regard  to  the  sun  when 
there  is  but  one  day  on  the  surface  of  the  earth? 

What  is  the  position  of  the  earth  when  one-half  of  the  earth's 
surface  has  one  day,  and  the  other  half  the  succeeding  day  ? 

What  is  the  position  of  the  earth  when  one-fourth  the  earth's 
surface  has  one  day,  and  the  remaining  three-fourths  has  the  suc- 
ceeding day? 

What  is  the  position  of  the  earth  when  one-fourth  the  earth's 
surface  has  one  day,  and  the  remaining  three-fourths  has  the 
preceding  day? 

At  what  time  of  day  for  us  does  the  succeeding  day  begin  at 
its  place  of  beginning? 

In  what  direction  does  the  earth  revolve? 

How  far  does  the  earth  revolve  in  one  hour?  In  one  minute? 
In  one  second? 

Do  places  east  of  us  have  earlier,  or  later,  time  than  we? 

At  2  o'clock  A.  M.  of  Monday  for  us,  is  it  Sunday,  or  Tuesday, 
in  some  other  part  of  the  world?  At  10  o'clock  A.  M.?  At  3 
o'clock  p.  M.? 

The  longitude  of  A  is  175  degrees  and  25  minutes  west,  and  of 
B  is  28  degrees  and  40  minutes  west;  what  is  the  time  at  B, 
when  it  is  3  hours  and  25  minutes  p.  M.  at  A? 

The  longitude  of  A  is  45  degrees  and  30  minutes  east;  what  is 
the  longitude  of  B,  whose  time  is  6  hours  50  minutes  A.  M.  when 
it  is  8  hours  and  20  minutes  p.  M.  at  A? 


THE  METRIC  SYSTEM. 


THE  METRIC  SYSTEM  OF  WEIGHTS 
AND  MEASURES. 


"In  whatever  it  is  our  duty  to  act,  those  matters  also  it  is  our 
duty  to  study." — Dr.  Arnold. 


The  Metric  System  of  Weights  and  Measures  has  been 
adopted  by,  and  is  used  in  a  modified  or  unmodified  form  in, 
France,  Spain,  Italy,  Greece,  Holland,  Germany,  Austria,  Nor- 
way and  Sweden,  Denmark.  Switzerland,  Portugal,  Belgium, 
Turkey,  Mexico,  Central  America,  Brazil,  Ecuador,  Peru,  Chili, 
Venezuela,  United  States  of  Columbia,  Argentine  Republic, 
Uruguay,  Paraguay,  Haiti,  Congo  Free  Stats,  British  India, 
Mauritius,  and  others;  while  the  United  States  of  America  and 
Great  Britain  have  authorized  its  use  within  their  respective  pos- 
sessions. Its  use  within  the  last  two  countries  is,  however,  at 
present,  very  limited  indeed.  The  merits  of  the  system  are 
recognized,  nevertheless,  by  the  citizens  of  both  the  United 
States  and  Great  Britain,  or  rather,  by  those  who  have  thorough- 
ly investigated  the  subject  of  exchange.  It,  therefore,  becomes 
the  duty  of  all  who  are  connected  with  educational  matters  and 
institutions  to  become  masters  of  this  subject,  to  the  end  that  the 
schools  may  become  the  medium  through  which  this  nation 
shall  advance  to,  and  take  her  place  in,  the  front  rank  on  this 
subject  also. 

Much  discretion  is  required  in  the  presentation  of  this  subject;  for 
the  aim  should  be  not  alone  to  teach  the  subject,  but  to  impress 
upon  the  pupil's  mind  the  beauties  and  advantages  of  the  system. 
Hence  the  custom  of  changing  from  the  units  of  one  system 
to  those  of  the  other  should  not  be  followed  at  all.  All  work 
should  be  done  wholly  within  the  Metric  System,  or  within  the 


134  THE  DEMAND. 

common  system,  then  the  pupil  will  readily  draw  correct  con- 
clusions as  to  the  comparative  merits  of  the  two.  If  the  subject 
is  properly  presented,  the  student  will  see  that  all  operations, 
reduction  ascending"  and  descending,  reduction  from  capacity  in 
liquid  units  to  capacity  in  solid  units,  to  weight  even,  under  cir- 
cumscribed circumstances,  is  a  matter  simply  of  removal  of  the 
decimal  point;  that,  as  a  system  of  weights  and  measures,  when 
compared  with  the  common  system,  it  is  analogous,  in  its  simple 
decimal  units,  to  our  money  system,  as  compared  with  that  of 
Great  Britain,  which  we  inherited,  disliked,  and  wisely  laid  aside. 
As  the  Metric  System  has  generally  been  taught,  if  taught  at  all, 
(that  is,  by  reducing  gallons  to  liters,  steres  to  cords,  miles  to 
meters,  and  the  like)  the  pupil,  and  the  teacher  even,  is  uncon  - 
sciously  led  to  the  false  conclusion,  that  the  processes  are  com- 
plicated, and,  therefore,  that  the  system  is  a  failure.  The  teacher 
forgets,  if  he  has  ever  thought,  that  the  business  man  will  have 
no  occasion  to  revert  to  the  old  system  if  the  new  is  employed; 
he  forgets  that  the  only  changes  which  will  be  needed,  or  ever 
used,  will  be  changes  from  meters  to  kilometers,  from  liters  to 
centiliters,  from  decigrams  to  quintals,  and  the  like;  and  that  all 
these  changes  can  be  made  without  any  other  process  than  that 
of  removing  the  decimal  point.  He  does  not  realize  that 
39.37079+  inches  will  be  perfectly  useless  to  him  then;  that 
35.316+  cubic  feet  will  not  be  needed;  that  .908+  quarts,  and 
all  the  other  terrible  numbers,  will  be  of  less  value  to  him  than 
the  hieroglyphics  on  Cleopatra's  needle.  Then,  let  the  following 
tables,  substantially  as  herein  given,  be  all  that  is  taught  or 
learned  from  the  text;  and  let  all  processes  be  wholly  within  the 
system,  and  consist  in  the  removal  of  the  decimal  point  only, 
whenever  possible;  and  that  is  impossible  only  when  computing 
square  and  cubical  contents,  having  the  linear  units  given. 

TABLE  OF  LENGTHS. 

10  Millimeters    —\  Centimeter. 
10  Centimeters  =1  Decimeter. 
10  Decimeters    =1  Meter. 
10  Meters  —1  Dekameter. 

10  Dekameters  =1  Hectometer. 
10  Hectometers =1  Kilometer. 
10  Kilometers    —1  Myriameter. 


THE  METRIC  SYSTEM.  135 

TABLE  OF  SURFACES. 


centiare. 


100  square  millimeters  =1  square  centimeter. 
100  square  centimeters  =1  square  decimeter. 

100  square  decimeters   =1  square  meter,  or  1  ct 

100  square  meters          =1  square  dekameter,  or  1  are. 
100  square  dekameters  =1  square  hectometer,  or  1  hectare. 
100  square  hectometers =1  square  kilometer. 
100  square  kilometers    =1  square  myriameter. 

TABLE  OF  SOLIDS. 

1000  cubic  millimeters  =1  cubic  centimeter. 

1000  cubic  centimeters  =1  cubic  decimeter. 

1000  cubic  decimeters    =  1  cubic  meter,  or  1  stere. 

1000  cubic  meters          —1  cubic  dekameter. 

1000  cubic  dekameters  —1  cubic  hectometer. 

1000  cubic  hectometers =1  cubic  kilometer. 

1000  cubic  kilometers    =1  cubic  myriameter. 

TABLE   OF  CAPACITY. 

10  milliliters    =1  centiliter. 

10  centiliters  =1  deciliter. 

10  deciliters    =1  liter. 

10  liters  =1  dekaliter. 

10  dekaliters  =1  hectoliter. 

10  hectoliters  =1  kiloliter. 

10  kiloliters     =1  myrialiter. 

TABLE  OF  WEIGHT. 

10  milligrams  =1  centigram. 

10  centigrams  =1  decigram. 

10  decigrams  =1  gram. 

10  grams  =1  dekagram. 

10  dekagrams  =1  hectogram. 

10  hectograms  =1  kilogram. 

10  kilograms  =1  myriagram. 


j.u  Kilograms  =1  mynagr 
10  myriagrams  — 1  quintal. 
10  quintals  =1  ton. 


\m  THE  DEMAND. 

EQUIVALENTS. 

1  stere=l  cubic  meter         =1  kiloliter  =1  ton  of  pure  water. 

1  cubic  decimeter  =1  liter        =1  kilogram  of  pure  water. 
I  cubic  centimeter  =  1  milliliter  =  l  gram  of  pure  water. 
1  cubic  millimeter  =1  milligram  of  pure  water. 

Actual  weights  and  measures  should  be  in  the  class  room 
whenever  possible. 

For  all  practicable  purposes,  there  are  but  three  names 
of  units  and  seven  prefixes  to  be  committed  to  memory,  in  all, 
ten  words  only.  The  task  of  learning  the  tables  is,  therefore,  a 
very  light  one;  for  the  numbers  require  no  attention  worthy 
of  mention. 


THE  PUBLIC  LANDS.  137 


THE  PUBLIC  LANDS. 


In  the  original  Thirteen  States,  the  Public  Lands  belonged 
originally  to  the  respective  states  in  which  they  are  situated;  and 
they  were,  by  the  several  states,  sold  or  otherwise  disposed  of  in 
accordance  with  the  laws  of  those  states.  Each  state,  therefore, 
has  a  system  of  subdivisions  of  the  lands  differing  more  or  less 
from  that  of  each  of  the  other  of  the  original  Thirteen  States; 
and,  frequently,  in  different  parts  of  the  same  state,  widely 
different  methods  of  subdividing  the  lands  obtained. 

Texas,  having  been  a  separate  Republic,  at  the  time  of  her 
annexation,  retained  full  control  of  the  Public  Lands  within  her 
boundaries.  Her  system  of  subdivisions  of  the  lands  is,  however, 
analogous  to  that  of  the  United  States. 

The  states  made  from  portions  of  the  original  Thirteen  States 
also  retained  control  of  the  Public  Lands  within  their  respective 
limits. 

In  all  other  portions  of  the  United  States,  the  Public  Lands 
belonged  to  the  United  States  Government;  and  ownership  to 
the  same  has  been,  or  must  be,  acquired  under  the  laws  of  the 
United  States. 

The  United  States  Government  has  established  a  uniform 
system  of  surveys  of  the  Public  Lands,  which  is  both  simple  and 
complete.  It  consists  in  dividing  and  subdividing  the  entire 
domain,  by  north  and  south  and  east  and  west  lines,  into  town- 
ships, as  nearly  as  may  be,  six  miles  square;  and  these  townships 
are  divided  into  sections  one  mile  square,  or  as  nearly  as 
may  be.  The  following  plats  and  descriptions  will  more  minutely 
represent  the  scope  and  plan  of  the  system : 


138 


THE  DEMAND. 

TOWNSHIPS. 


Stan? 

lard 

Pan 

illel. 

TWTE5E 

T5HE4W 

O) 

1 

3 

T3NB5W 

T3NE2E 

Base 

T1NE1W 
Ini 

d 

T1NE1E 

tial 
fe 

Line. 

=5? 

Poi 

T1SE1W 

«• 

nt. 

T1SE1E 

T2SS5E 

T3SB4W 

C 
.5 

'S 

T4S32E 

Stan 

dard 

<D 
S 

Para 

llel. 

X6U3W] 

In  the  surveys  of  the  Public  Lands  of  the  United  States, 
townships  are  six  miles  square,  or  as  nearly  as  is  possible,  con- 
sidering that  their  exterior  boundaries  are  north  and  south  and 
east  and  west  lines. 

Some  prominent  natural  point  is  usually  chosen  as  an  initial 
point  through  which  the  base  line  and  the  meridian  line  pass. 
These  initial  points  are  established  at  convenient  locations,  so 
that  they  may  be  centers  of  extensive  areas,  including  sometimes 
more  and  sometimes  less  than  the  respective  States  or  Territories 
in  which  they  are  situated.  For  instance,  in  California  there  are 
two  initial  points,  Mount  Diablo  and  Mount  San  Bernardino,  the 
system  of  each  of  which  extends  very  far  beyond  the  bound- 


THE  PUBLIC  LANDS. 


139 


aries  of  California.  At  intervals  of  about  thirty  miles  north,  and 
south,  of  the  base  line,  Standard  Parallels  are  established  for  the 
purpose  of  correcting  the  errors  arising  from  the  convergency  or 
divergency  of  the  meridian  lines.  On  these  Standard  Parallels, 
there  is  a  double  set  of  corners. 

SECTIONS  OF  A  TOWNSHIP. 


/ 

1 

i 

m 

t 

[    ; 

i 

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1 

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7 

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6 

Sections  are  one  mile  square,  or  one  thirty-sixth  of  a  township, 
as  nearly  as  may  be,  considering  the  convergency  of  meridians, 
and  errors  otherwise  inseparable  from  the  system  of  surveys. 
All  errors,  excesses,  and  deficiencies,  from  whatever  cause,  are 
crowded  against  the  northern  and  western  boundaries  of  the 
township.  Hence,  all  sections,  except  those  against  the  northern 
and  western  boundaries  of  the  township,  contain  exactly  640 
acres,  while  those  excepted  usually  contain  either  less  or  more 
than  640  acres.  Sections  one,  two,  three,  four,  five,  seven, 
eighteen,  nineteen,  thirty,  and  thirty-one,  are  so  divided  that  they, 
severally  contain  two  full  quarter-sections  (160  acres)  and  two 
full  eighties;  while  section  six  is  so  divided  that  it  contains  one 
full  quarter-section,  two  full  eighties,  and  one  full  forty-acre  lot. 


14(1 


THE  DEMAND. 


The  remainders  of  these  sections  are  divided  into  lots  containing 
somewhat  more  or  less  than  forty  acres,  and  are  accounted  to 
contain  neither  more  nor  less  than  their  true  acreage. 


^ 


40±        40±        40± 


100  A 


«0  A 


ICO  A 


•£ 


'i 


160  A 


160  A 


160  A 


160  A 


THE  P  UBLIC  LANDS.  141 

Corners  are  established  every  half-mile  on  the  exterior  bound- 
aries of  townships  and  sections,  as  shown  in  the  diagrams,  and 
whenever  practicable,  they  are  marked  as  therein  shown.  In 
addition  to  the  marking  of  the  post  which  is  set  as  shown  in  the 
diagrams,  one  tree  at  each  post  and  in. each  section  is  marked  BT, 
together  with  the  name  of  the  township  and  section  in  which 
it  is  situated;  and  its  direction  and  distance  from  the  corner  is 
recorded  in  the  field-notes.  Any  tree  that  is  exactly  on  the  line 
is  marked  with  two  notches,  or  chops,  on  each  of  its  opposite 
two  sides  in  the  directions  of  the  line.  Trees  near  the  line  are 
blazed  twice,  once  facing  the  direction  whence  the  line  was 
surveyed,  and  once  facing  the  direction  whither  it  was  surveyed, 
both  blazes  being  nearer  the  side  of  the  tree  facing  the  line  at 
right  angles,  than  the  one  opposite  the  line. 

The  foregoing  are  the  methods  employed  in  surveying  in 
timbered  localities. 

In  stony  localities,  a  monument  of  stones  is  used  for  a  township 
corner,  and  a  single  stone  for  a  section  and  a  quarter-section 
corner. 

All  posts  and  stones  used  as  corners  are  notched  as  follows, 
on  the  edges  facing  the  cardinal  points  of  the  compass :  Township 
corners,  six  notches  on  each  edge;  all  other  section  corners  are 
notched  with  as  many  notches  on  their  respective  edges  as  will 
indicate  the  number  of  miles  the  corner  is  situated  from  the 
township  boundary  which  the  edge  faces. 

In  portions  of  the  country  in  which  there  is  neither  stone  nor 
timber,  mounds  of  earth  are  thrown  up  at  the  section  corners  in 
a  conical  shape,  and,  surrounding  each  mound,  a  trench  in  the 
form  of  a  square,  with  its  corners  on  the  line.  The  trenches  at 
the  half-mile  corners  are  of  the  same  shape,  but  the  sides  of  the 
square  are  parallel  with,  and  p'erpendicular  to,  the  section  lines. 
Mounds  of  earth  or  stone  are  built,  also,  where  there  are  no 
bearing  trees,  even  if  there  are  marked  posts  or  stones.  Within 
the  mound  is  deposited  a  stone  or  some  charcoal.  The  form 
of  the  mounds,  trenches,  and  pits,  which  are  inseparable,  are  as 
shown  in  the  annexed  diagrams: 


142 


THE  DEMAND. 


o 


PIT 


PIT 


o 


o 


PIT 


PIT 


o 


PIT 


PIT 


THE  PUBLIC  LANDS.  148 

TESTS. 

Draw  a  plot  of  a  section  of  land,  and  locate  the  N.  E.  ^,  and 
the  N.  \  of  S.  W.  ^ 

Draw  a  plot  of  a  township  of  land,  and  locate  the  N.  W.  \  of 
the  S.  E.  \  of  section  9.  How  many  acres  are  there  in  the 
location? 

Draw  a  plot  of  a  township  of  land;  subdivide  it  and  locate  the 
S.  W.  \  of  section  11,  the  S.  |-  of  section  10,  and  the  W.  | 
of  the  N.  W.  \  of  section  15.  What  is  the  land  located  worth  at 
$25.  per  acre,  and  what  will  it  cost  to  fence  it  at  $1  per  rod  ? 

A  man  in  New  York  City,  owning  a  half  section  of  land, 
requested  me  to  sell  it  for  him  at  $65  per  acre,  on  a  commission 
of  one  and  one-half  per  cent,  and  to  remit  the  proceeds  to  him  in 
a  60-day  draft;  exchange  being  at  2  per  cent  premium,  and 
interest  6  per  cent,  what  was  the  face  of  the  draft? 

The  man  who  bought  the  land  mentioned  in  the  preceding 
problem  wished  me  to  have  it  fenced  for  him  at  $1.50  per  rod, 
he  allowing  me  a  commission  of  three  per  cent;  and  he  then 
wished  me  to  sell  the  land  for  him  at  a  net  profit  of  10  per  cent, 
allowing  me  a  commission  of  1  per  cent  for  selling;  for  how 
much  was  the  land  sold? 


144  THE  DEMAND. 


AVERAGE   OF  ACCOUNTS. 


This  subject  is  of  immediate  interest  to  only  a  small  portion 
of  the  people,  and  therefore  should  occupy  a  secondary  position. 
The  purpose  of  averaging  accounts  is  to  determine  the  date  from 
which  interest  shall  be  computed  on  the  balance  of  the  account, 
so  that  there  shall  be  no  loss  to  either  party;  or,  to  ascertain  the 
date  of  a  note  given  in  settlement,  and  whose  face  is  the 
balance.  This  subject  requires  but  little  outside  investigation, 
as  all  the  ordinary  business  transactions  leading  up  to  it  are 
already  familiar  to  the  pupils.  The  methods  of  computing,  and 
the  interpretations  of  the  restflts,  when  the  date  from  which 
interest  is  computed  falls  either  before  all  the  dates  of  the 
transaction,  or  after  all  the  dates  of  the  transaction,  constitute 
the  important  work  under  this  subject.  In  other  words,  the 
interest  of  the  student  will  center  in  the  methods  of  procedure 
and  the  results.  DeGarmo  says:  "The  teacher  is  needed  for 
the  steps  which  the  children  cannot  take  alone,  the  derivations 
and  applications  which  they  would  not  or  could  not  make;  con- 
sequently, instruction  should  deliberately  plan  for  these  greater 
matters  of  education,  leaving  the  small  ones  to  an  awakened 
spontaneity  of  the  pupil,  or  to  incidental  instruction." 

CLASS   WORK. 

Problem:  The  following  account  was  settled  by  the  debtor 
giving  his  note  for  the  balance.  Determine  the  face,  date,  and 
maker  of  the  note. 


A  VERAGE  OF  ACCO  UNTS. 


14o 


Dr.  A.  B. 

1890. 

May    6  Mdse  $7150. 

16      "  475. 

June  17      "  3475.25 

21      "  1516.50 

July     5      "  279.90 

Dr. 

Solution:   $7150.   x  0-     0. 
475   xlO=  4750. 
3475.25x42  =  145960.5 
1516.50x46-  69759. 
279.90x60=  16794. 
$12896.65     237263.5 


Cr. 
1890. 

May  9  Cash  $2450. 
21   "    915. 
June  12  Mdse  4165.50 
19   "   2915.50 


3475.25 
42 


Cr. 

$2450.   x  3  = 
915   x!5  = 


7350. 
13725. 


4165.50x37=  154123.5 
2915.50x44=  128282. 
$10446.        303480.5 


2450.65   ) 

66217.00 
49013  0 

(27 

2915.5 
44 

17204  00 
17154  55 

116620 
116620 

128282 


695050 
1390100 
145960.50 


1516.5 

46_ 

90990 
60660 
69759 
4165.5 

37 

291585 
124965 
154123.5 


May  6 — 27  days = April  9  =  date  of  note. 

$2450. 65= face  of  note. 

A.   B.  =the  maker  of  the  note. 

Explanation:  The  debit  side  of  the  account  represents  what 
A  B  owes  to  C  D,  and  the  credit  side  of  the  account  represents 
what  C  D  owes  to  A  B.  The  debit  side  being  $2450.65  greater 
than  the  credit  side,  the  face  of  the  note  must  be  $2450.65  and  the 
maker  A  B.  The  date  of  the  note  is  yet  to  be  determined.  If 
the  note  were  dated  May  6,  1890,  A  B  would  neither  gain  nor 
lose  any  interest  on  the  $7150,  since  that  amount  of  merchandise 
was  bought  on  that  day.  He  would  however  lose  the  interest 


146  THE   DEMAND. 

of  $475  for  ten  days,  or  the  interest  of  $1  for  4750  days;  he 
would  lose  the  interest  of  $3475.25  for  *  42  days,  or  the  interest' 
of  $1  for  145,960.5  days;  also  the  interest  of  $1516.50  for  46 
days,  or  the  interest  of  $1  for  69,759  days;  and  the  interest  of 
$279.90  for  60  days,  or  the  interest  of  $1  for  16,794  days.  He 
would  lose,  therefore,  the  interest  of  $1  for  237,263.5  days,  on 
the  debit  items,  if  the  note  were  dated  May  6,  1890. 

He  would,  however,  gain  the  interest  of  $2450  for  three  days, 
or  the  interest  of  $1  for  7,350  days;  the  interest  of  $915  for  15 
days,  or  the  interest  of  $1  for  13,725  days;  the  interest  of 
$4,165.50  for  37  days,  or  the  interest  ol  $1  for  154,123.5  days; 
and  the  interest  of  $2915.50  for  44  days,  or  the  interest  of  $1 
for  128,282  days.  He  would  gain,  therefore,  the  interest  of  $1 
for  303,480.5  days,  on  the  debit  items;  he  would  gain  on  the 
whole  account  the  interest  of  $1  for  66,217  days;  but  the  interest 
of  $1  for  66,217  days  is  equal  to  the  interest  of  $2450.65  for  27 
days.  Hence  the  note  must  be  dated  27  days  anterior  to  May 
6,  or  April  9,  1890. 


PROPORTION.  147 


PROPORTION. 


Since  all  problems  that  can  be  solved  arithmetically  by  pro- 
portion can  be  solved  quite  readily  under  other  principles,  it 
seems  that  proportion,  from  an  arithmetical  stand-point,  is  more 
valuable  as  a  device  than  as  a  principle.  If  it  is  to  be  considered 
as  a  device,  it  should  be  employed  in  such  way  as  to  avoid  as 
much  mental  and  physical  labor  as  possible.  In  other  words, 
it  should  be  as  simple  and  direct  as  possible.  By  the  nature  of 
business  transactions  it  is  evident  that  cost  and  quantity,  time  of 
labor  and  labor,  etc.,  under  given  conditions  'bear  the  same 
relations  to  other  like  quantities  under  the  same  conditions; 
hence  proportion  is  simply  a  comparison  of  the  elements  of  two 
transactions,  in  one  of  which  the  elements  are  all  known,  and  in 
the  other  of  which  one  element  is  unknown.  The  method  is 
clearly  outlined  in  the  solutions  and  explanations  of  the  following 
problems : 

Problem:  If  a  garrison  of  3600  men  have  bread  enough  to 
last  them  thirty-five  days,  if  each  be  allowed  24  ounces  per  day, 
how  many  men,  at  14  ounces  per  day  each,  will  require  twice  as 
much  for  45  days? 

Solution:  3600  ? 

35     :     45     ::     1     :     2 
•  24          14 

400 

2  X  24  x  -35  x  3600  men =9600  men. 


Explanation :  It  is  evident  that  the  ratio  which  3600  times  35 


148  THE  DEMAND. 

times  24  ounces  bear  to  ?  times  45  times  14  ounces  is  as  one  is 
to  2.  It  has  already  been  established  that,  In  any  proportion 
the  product  of  the  means  equals  the  product  of  the  extremes. 
0600  is  the  number  that  will  produce  this  equality;  hence  9600 
men  is  the  answer  to  the  question. 

Problem:  If  82  men  build  a  wall  36  feet  long,  8  feet  high,  and 
4  feet  thick,  in  4  days,  in  what  time  will  48  men  build  a  wall  864 
feet  long,  6  feet  high,  and  3  feet  wide? 

Solution:  82     :     48     ::     36     :     864 

4  ?  8  6 

4  3 

9 

ft*    41 
$X0x804x82x4days  =  369=92|  days. 

4x0x$$x$*  4 

0       4 

Explanation:  It  is  evident  that  the  ratio  that  82  times  4  days 
bears  to  48  times  ?  days  is  equal  to  the  ratio  that  36  times  8 
times  4  cubic  feet,  built  by  the  82  men  in  4  days,  bears  to  the  864 
times  6  times  3  cubic  feet,  built  by  the  48  men  in  the  required 
number  of  days;  for  the  quantity  of  work  performed  always  has 
a  given  relationship  to  the  number  of  days  required  to  perform 
it,  all  other  conditions  being  equal.  It  has  already  been 
established  that,  In  any  proportion  the  product  of  the  extremes 
equals  the  product  of  the  means.  92^  is  the  number  that  will 
produce  this  equality;  therefore,  92^  days  is  the  time  required 
for  the  48  men  to  work. 


GENERAL  AVERAGE—  SHIPPIXU.  149 


GENERAL  AVERAGE— SHIPPING. 


<(The  aim  of  education  must  be  to  arouse  in  the  pupil  this 
spiritual  and  ethical  sensitiveness  which  does  not  look  upon  any- 
thing as  merely  indifferent,  but  rather  knows  how  to  seize  in 
everything,  even  in  the  seemingly  unimportant,  its  universal 
significance. " — Rosencranz. 


This  is  one  of  the  subjects  that  directly  affect  such  a  small 
portion  of  the  population  that  its  treatment  should  be  reserved 
until  the  subjects  of  more  nearly  general  application  are  thor- 
oughly comprehended.  In  seaboard  towns  and  cities,  it  will  be  a 
more  vital  subject  than  in  inland  localities  where  the  shipping 
interests  are  a  minimum.  Hence,  its  consideration  should  cover 
a  greater  period  of  time,  be  much  more  exhaustive  in  its  nature, 
and,  perhaps,  be  undertaken  at  an  earlier  age  by  a  maritime 
people  than  by  others.  This  may  be  stated  as  a  general  prin- 
ciple, that  subjects  should  receive  attention  in  accordance  with 
the  vocations  of  the  people.  Students,  in  starting  out  to  investi- 
gate the  subject  of  General  Average,  should  be  given  an  outline 
of  points  to  be  their  guide,  so  that  random  work  may  be  avoided 
and  valuable  time  saved. 

.  OUTLINE. 

1.  Interview  ship  owners,  ship  captains,  and  shippers. 

2.  Ascertain  the  value  of  ships  and  cargoes. 

3.  Ascertain  the  time  and  expenses  of  voyages. 

4.  Ascertain  the  rates  of  freightage. 

5.  Review  maritime  insurance. 

6.  Ascertain  the  frequency  and  extent  of  disasters. 

7.  Ascertain  the  probable  duration  of  ships. 


150  THE  DEMAND. 

8.  Ascertain  the  mode  of  procedure  in  cases  of  imminent  peril, 

9.  Ascertain  how  losses  are  sustained;    whether  by  persons 
immediately  affected  or  by  all,  and,  if  by  all,  in  what  ratio. 

Out  of  the  vast  amount  of  material  gathered,  the  skillful 
teacher  can  select,  classify,  and  retain,  as  working  power,  for  the 
pupils'  future  guidance,  the  essentials  only.  The  matter  of  com- 
merce however  after  this  investigation  is  to  the  pupil  no  longer  a 
terra  incognita,  but  a  discovered  and  active  realm,  affecting  the 
well  being  of  the  entire  civilized  world.  The  possibilities  of  man 
are,  in  the  minds  of  the  pupils,  greater  and  greater  every  day 
of  their  lives.  They  are  more  and  more  impressed  with  the  idea 
that  they  must  do  and  become  more  and  more  as  the  days  pass, 
if  they  would  attain  anything  above  mediocrity.  The  sluggard 
or  the  dullard,  they  are  convinced,  has  no  place  in  such  an  active 
world.  Onward  and  upward  is  the  only  course  that  will  adorn 
and  honor.  Teaching  is  a  pleasure  when  the  pupils  are  imbued 
with  these  ideas.  Out  of  abundant  desire  fruitful  results  are 
sure  to  grow. 


ALLIGATION. 


Alligation  is  another  subject  whose  importance  is  restricted, 
owing  to  the  fact  that  few  employ  its  ordinary  methods  in  any 
business  transactions.  When  the  quantities  and  price  per  unit 
of  quantity  are  given,  and  the  average  price,  without  gain  or 
loss,  is  required,  the  process  is  so  simple  as  to  need  no  explana- 
tion. The  phase  of  the  subject  that  calls  for  thoughtful  con- 
sideration is  the  one  in  which  the  average  price  and  the  several 
prices  per  unit  are  given,  to  find  sets  of  quantities  that  will  satisfy 
these  conditions,  without  gain  or  loss.  Sometimes  certain  in- 
gredients are  to  be  used  in  specified  amounts  in  order  to  work 
off  a  certain  kind  of  stock,  and  sometimes  the  entire  mixture  is 
to  be  of  a  specified  quantity  in  order  to  fill  a  certain  vessel,  a 
certain  order,  or  to  fulfill  any  other  certain  specified  conditions ; 
then  the  problem  becomes  severer. 


MENTAL  DISCIPLINE,  151 


MENTAL  DISCIPLINE. 


"Repeated  attacks,  by  concentrated  attention,  not  only  master 
the  abstruse  problem,  but  leave  the  mind  with  a  permanent 
acquisition  of  power  of  analysis  for  new  problems." — Harris. 


Problem:  A  can  do  a  piece  of  work  in  8  days,  and  B  can  do 
the  same  work  in  9  days;  in  what  time  can  both  do  the  same 
work,  working  together? 

Solution:  |+i=H-          H+tt=±&  days. 

Explanation:  Since  A  can  do  the  work  in  8  days,  in  1  day  he 
can  do  one-eighth  of  the  work;  and  since  B  can  do  it  in  9  days, 
in  one  day  he  can  do  one-ninth  of  the  work;  both,  therefore,  in 
one  day  can  do  the  sum  of  one-eighth  and  one-ninth  of  the 
work,  or  seventeen  seventy-seconds  of  the  work;  and  to  do 
seventy-two  seventy-seconds,  or  the  whole  work,  will  require  as 
many  times  one  day  as  seventeen  seventy-seconds  is  contained 
times  in  seventy-two  seventy-seconds,  or  four  and  four-seven- 
teenths times  one  day,  or  four  and  four-seventeenths  days. 

Problem  :  A  and  B  can  do  a  piece  of  work  in  14  days,  and  A 
can  do  six-sevenths  as  much  as  B;  in  what  time  can  each  do  the 
work  alone? 


Solution  :  Y  of  14  =  30i  daYs.     A- 

ty  of  14  =26  days,     B. 

Explanation:    Since  A  can  do  six  -sevenths  as  much  as  B  in  a 
given  time,    both  can  do  thirteen-sevenths  as  much  as  B   in   a 


152  THE  DEMAND. 

given  time.  Since  A  was  14  days  doing  six-sevenths  as  much  as 
B  did  in  14  days,  to  do  one-seventh  as  much  would  require  one- 
sixth  of  14  days,  and  to  do  thirteen-sevenths  as  much,  or  what 
both  did,  would  require  thirteen  times  one-sixth  of  14  days,  or 
30$  days. 

Since  B  was  14  days  doing  seven-sevenths  as  much  as  he  did 
in  fourteen  days,  to  do  one-seventh  as  much  would  require 
one-seventh  of  14  days,  or  2  days;  and  to  do  thirteen-sevenths 
as  much,  or  what  both  did,  would  require  thirteen  times  2  days, 
or  26  days. 

Problem:  A  and  B  can  do  a  piece  of  work  in  8  days;  A  and  C 
can  do  the  same  piece  of  work  in  9  days;  and  B  and  C  in  6 
days.  In  what  time  can  all  do  the  work,  working  together?  In 
what  time  can  each  do  the  work  alone? 

Qr»1nfir>n  •      1  _l_  1  _J_  1  _  29       29    .    9__29_       144    .      29  __  A  28    rlotrc     oil 

on-    fTT'TT^rffj    TT~^  —  TTT>    T¥¥~TT7  —  42~9  days,  all. 


29  _  1  .  _  11    144  .   11  _ 

T¥T   f  —  T4"T'  lf¥~    — 


days,  B. 
-T^=28i  days,  A. 


Explanation:  Since  A  and  B  can  do  the  work  in  8  days,  in  1 
day  they  can  do  one-eighth  of  the  work;  and  likewise  A  and  C 
in  1  day  can  do  one-ninth  of  the  work,  and  B  and  C  in  1  day 
can  do  one-sixth  of  the  work.  Then  twice  what  A,  B,  and  C 
can  do  in  one  day  is  the  sum  of  one-eighth,  one-ninth,  and 
one-sixth  of  the  work,  or  twenty-nine  seventy-seconds  of  the 
work;  hence  what  they  can  do  in  1  day  is  one-half  of  twenty- 
nine  seventy-seconds,  or  twenty-nine  one-hundred-forty-fourths, 
of  the  work.  They  can  do  the  entire  work,  therefore,  in  as 
many  days  as  twenty-nine  one-hundred-forty-fourths  is  con- 
tained times  in  one-hundred-forty-four  one-hundred-forty-fourths, 
or  four  and  twenty-eight  twenty-ninths  times  one  day,  or  four  and 
twenty-eight  twenty-ninths  days.  All  can  do  twenty-nine  one 
hundred-forty-fourths  of  the  work  in  one  day;  A  and  B  can  do 
one-eighth  of  the  work  in  one  day;  C,  therefore,  can  do  the 
difference  between  twenty-nine  one-hundred-forty-fourths  and 


MENTAL  DISCIPLINE.  153 

one-eighth,  or  eleven  one-hundred-forty-fourths  of  the  work  in 
one  day.  He  can  do  the  entire  work  in  thirteen  and  one- 
eleventh  days.  Likewise,  B  can  do  it  in  eleven  and  one-thirteenth 
days,  and  A  in  twenty-eight  and  four-fifths  days. 

Problem :    At  what  time  between  two  and  three  are  the  hands 
of  a  clock  twenty  minute-spaces  apart? 


Solution:.  yfx 30=32T8T  minutes  past  2. 

and 
|2.  x50=5415r  minutes  past  2. 


Explanation:  Since  the  minute  hand  passes  entirely  around 
the  face  of  the  clock  while  the  hour  hand  passes  over  a  five- 
minute-space,  the  minute  hand  gains  on  the  hour  hand  eleven 
minute-spaces  in  going  over  twelve  minute-spaces.  At  two 
o'clock  the  hands  are  ten  minute-spaces  apart;  it  is  evident, 
therefore,  that  the  minute  hand  must  gain  ten  minute-spaces 
to  over-take  the  hour  hand,  and  must  gain  twenty  minute-spaces 
more  to  be  twenty  minute-spaces  beyond  it.  Hence  the  minute 
hand  must  gain  in  all  thirty  minute-spaces;  and  as  it  gains 
eleven  spaces  in  going  twelve,  it  will  go  twelve-elevenths  of 
thirty  minutes -spaces  after  two  o'clock.  Hence,  at  thirty-two 
and  eight-elevenths  minutes  past  two  o'clock,  the  hands  are 
twenty  minute-spaces  apart.  They  are  twenty  minute-spaces 
apart  again  at  fifty-four  and  six-elevenths  minutes  past  two 
o'clock. 

Problem:    What  is  the  time  when  the  minute  hand  lacks  as 
much  of  being  at  the  IV-mark  as  the  honr  hand  is  beyond  it? 


154 


Solution: 


THE  DEMAND. 
of  20 =18&  past  4. 


Explanation:  The  minute  hand  moves  twelve  times  as  fast  as 
the  hour  hand,  hence  when  the  hands  have  the  required  position 
the  minute  hand  will  have  gone  twelve  times  as  far  since  four 
o'clock  as  the  hour  hand,  and  since  it  is  then  as  far  back  of  the 
IV-mark  as  the  hour  hand  is  beyond  it,  it  is  clear  that  the 
distance  from  the  XI I -mark  to  the  IV-mark  is  thirteen  times 
the  distance  that  the  hour  hand  is  beyond  the  IV-mark;  the 
minute  hand  is  only  twelve-thirteenths  of  the  distance  to  the 
IV-mark;  if  it  were  at  the  IV-mark  the  time  would  be  twenty 
minutes  past  4  o'clock;  the  time,  therefore,  that  fulfills  the 
conditions  of  the  problem  is  twelve-thirteenths  of  twenty  minutes 
past  4  o'clock,  or  eighteen  and  six-thirteenths  minutes  past 
4  o'clock. 

Problem :  I  wish  to  pay  both  principal  and  interest  of  a  debt 
of  $4000  in  four  equal  payments  at  the  end  of  1,  2,  3,  and  4 
years,  respectively,  the  rate  of  interest  being  7  per  cent.  What 
shall  be  the  amount  of  each  payment?  How  much  of  each  pay- 
ment is  principal,  and  how  much  is  interest? 

Solution:     100%  =lst  payment  of  principal. 

107%  =2d  payment  of  principal. 

114.49%      =3d  payment  of  principal. 
122. 5043%=  4th  payment  of  principal. 
<<0.9943%  of  the  1st  payment  of  principal = the  4 
icinal.   or  #4000 


443.9943%  of  the  1 
payments  of  principal,   or  $4000. 


MENTAL  DISCIPLINE.  loo 

443. 9943)4000. 0000(9. 0091  =  1  % . 
3995  9487 
4  0513000 
3  9959487 


5535130 
4439943 

100%=$  900.91=lst  payment  of  principal. 
280.      =lst  payment  of  interest. 
1180. 91= amount  of  each  payment. 

4000  9.0091 

900.91  107 


3099.09  630637 

07  90091 


216.9363  $963.9737  2d  payment  of  principal. 

216.9363  2d  payment  of  interest. 
1180.91       amount  of  each  payment. 

Explanation:  During  the  first  year  the  entire  $4000  draws 
interest;  during  the  second  year,  a  portion  of  the  principal 
having  been  paid,  the  interest  is  less  than  the  interest  the  first 
year;  hence  more  principal  must  be  paid  to  keep  the  entire  pay- 
ments equal.  Likewise,  each  succeeding  year  the  interest  de- 
creases and  the  payment  of  principal  increases. 

The  payment  of  principal  the  first  year  is  one  hundred  per 
cent  of  itself.  The  second  year  no  interest  will  be  required  to  be 
paid  on  this  hundred  per  cent;  but  seven  per  cent  of  one  hundred 
per  cent,  or  seven  per  cent  of  the  first  payment  of  principal,  is 
the  amount  of  interest  this  payment  of  principal  bore  the  first 
year;  hence,  the  second  year  the  interest  will  be  less  than  the 
interest  the  first  year  by  seven  per  cent  of  the  first  payment 
of  principal;  hence,  to  make  the  entire  second  payment  equal 
the  entire  first  payment,  the  payment  of  principal  must  be 
increased  seven  per  cent  of  the  first  payment  of  principal;  hence 
the  second  payment  of  principal  must  be  one-hundred-seven 
per  cent  of  the  first  payment  of  principal.  By  the  same  course 
of  argument,  the  third  payment  of  principal  will  be  seven  per 
cent  of  the  second  payment  of  principal  more  than  the  second 
payment  of  principal,  and  the  fourth  payment  of  principal  seven 


156  THE  DEMAND. 

per  cent  of  the  third  more  than  the  third.  All  the  payments 
of  principal  are,  therefore,  443.9943%  of  the  first  payment  of 
principal;  but  all  the  payments  of  principal  are  $4000.  Hence 
443.9943%  of  the  first  payment  of  principal  equals  $4000;  one 
per  cent  equals  $9.0091;  and  one  hundred  per  cent,  or  the  first 
payment  of  principal,  equals  $900.91.  The  interest  on  $4000  for 
one  year  at  seven  per  cent  is  $280;  hence  the  entire  payment  the 
first  year,  and  consequently  every  year,  is  $1080.91.  The  pay- 
ment of  principal  the  second  year  is  107  X  $9. 0091;  the  third  year, 
114. 49  x  $9. 0091;  and  the  fourth  year,  122. 5043  x  $9. 0091.  The 
interest  the  second  year  is  seven  per  cent  of  $4000 =$900. 91; 
and  that  the  third  and  fourth  years  is  similarly  ascertained. 

Problem:  A  man  wishes  to  pay  principal  and  interest  of  a  debt 
of  $5000  in  three  equal  payments  in  0,  1,  and  2  years  from  date. 
If  the  debt  bear  8  per  cent  interest,  what  must  be  the  amount 
of  each  of  the  three  equal  payments  that  will  discharge  the  debt  ? 
How  much  of  each  payment  is  principal,  and  how  much  is 
interest  ? 

Solution: 

100%       =payment  of  principal  at  end  of  1  year. 
108%       =payment  of  principal  at  end  of  2  years. 
116.64%  =  payment  of  principal  at  end  of  0    years. 
324.64%  of  payment  of  principal  at  end  of  1st  year =$5000. 

324.64)5000.00(15.40167=1%  of  payment  of  principal  at  end 
3246  4  of  1st  year. 

1753  60 
1623  20 

130  400 

129  856 


219360 

194784 

24576 


100%  =$1540.17=2d  payment  of  principal. 
108%  =  1663. 38= 3d  payment  of  principal. 
116.64%—  1796.45=cash  pay't,  or  pay't  in  0  years. 


MENTAL  DISCIPLINE.  157 

5000 

1796.45 

3203.55 

.08 

$  256.284  interest  at  end  of  1st  year. 

1540.17 
$1796.45  entire  payment  at  end  of  1st  year. 

Explanation:  The  payment  in  0  years,  that  is,  the  cash  pay- 
ment, is  necessarily  all  principal,  as  no  interest  has  had  time 
to  accrue.  The  second  payment,  or  the  payment  at  the  end 
of  the  first  year,  contains  the  most  interest,  and,  therefore,  the ' 
smallest  payment  of  principal.  Then  by  the  same  course  of 
argument  employed  in  the  preceding  problem  the  second  pay- 
ment of  principal  will  be  100%  of  itself,  and  the  third  payment 
of  principal  108%  of  the  second  payment  of  principal.  The 
third  payment  of  principal  is  the  final  payment  of  principal  and, 
consequently,  the  amount  of  interest  paid  at  the  final  payment 
is  8  per  cent  of  108  per  cent  of  the  second  payment  of  principal. 
But  8  per  cent  of  108  per  cent,  added  to  108  per  cent,  equals 
116.64  per  cent,  which  is  the  entire  amount  paid  at  the  last 
payment.  The  entire  amount  paid  at  the  last  payment  equals 
the  entire  amonut  paid  at  any  other  payment;  consequently,  the 
cash  payment  must  be  116.64  per  cent  of  the  principal  paid  at  the 
second  payment.  Then  324.64  per  cent  equals  $5000;  and  the 
payment  of  principal  at  the  end  of  the  first  year  (second  pay- 
ment) equals  $1540.17;  at  the  end  of  the  second  year,  (last  pay- 
ment) $1663.38;  and  the  cash  payment,  (first  payment)  $1796.45. 
This  payment  was  all  cash,  hence  the  entire  payment  each  year 
was  $1796.45.  $5000— $1796.45,  or  $3203.55,  at  8  per  cent  for 
one  year  equals  $256.28,  the  interest  paid  at  the  end  of  the  first 
year.  $1540. 17 +$256. 23 =$1796.45,  the  entire  payment  again. 
The  princpal  to  bear  interest  the  second  year  is  the  third  pay- 
ment of  principal,  $1663.38,  which  at  8  per  cent  for  one  year 
produces  $133.07,  the  last  payment  of  interest.  $1664.38+ 
$133. 07  =$1796. 45  the  entire  payment  again. 

TESTS. 

A  and  B  can  do  a  piece  of  work  in  9  days;    A  and  C  can  do 


158  THE  DEMAND. 

the  same  work  in  12  days;  and  B  and  C,  in  15  days.  In  what 
time  can  each  do  the  piece  of  work  alone  ? 

A  and  B  can  do  a  piece  of  work  in  15  days;  A  can  do  five- 
sixths  as  much  as  B  in  a  given  time.  In  what  time  can  each  do 
the  piece  of  work  alone? 

When  will  the  hands  of  a  watch  be  together  between  two  and 
three  o'clock?  Opposite?  At  right  angles?  Five  minute-spaces 
apart?  Twenty  minute-spaces  apart? 

What  is  the  time  when  the  hour  hand  is  as  far  beyond  the 
Vl-mark  as  the  minute  hand  lacks  of  being  at  the  VUI-mark? 

What  is  the  time  when  the  minute  hand  is  twice  as  far  beyond 
the  II-mark  as  the  hour  hand  lacks  of  being  at  the  V-mark  ? 

Divide  $2000  among  A,  B,  and  C,  in  the  proportion  of  one- 
third,  one-fourth,  and  one-sixth. 

I  wish  to  pay  the  interest  and  principal  of  an  interest-bearing 
debt  of  $2000,  in  four  equal  annual  payments,  one,  two,  three, 
and  four  years  from  date.  The  rate  of  interest  which  the  debt 
bears  is  six  per  cent.  How  much  must  I  pay  at  each  payment  ? 
How  much  of  the  first  payment  is  principal,  and  how  much  is 
interest?  Of  the  fourth? 

I  wish  to  pay  the  interest  and  principal  of  an  interest-bearing 
debt  of  $4000,  in  four  equal  payments,  one  year  apart;  the  first 
in  cash,  the  second  at  the  end  of  one  year,  the  third  at  the  end 
of  two  years,  and  the  fourth  at  the  end  of  three  years.  The  rate 
of  interest  which  the  debt  bears  is  8  per  cent.  How  much  must 
I  pay  at  each  payment?  How  much  of  the  second  payment  is 
principal,  and  how  much  is  interest  ?  Of  the  third  ? 

A,  B,  and  C  can  do  a  piece  of  work  in  nine  days;  A  can  do 
three-fourths  as  much  as  B;  and  B  can  do  four-fifths  as  much  as 
C.  In  how  many  days  can  each  do  the  work  alone? 


RESUME.  159 


RESUME. 


"The  entire  succession  of  men,  through  the  whole  course  of 
ages,  must  be  regarded  as  one  man  always  living  and  constantly 
learning. " — Pascal. 


In  considering  the  grandeur  of  this  nation,  the  people  are 
considering  the  potency  of  the  public  school  system;  for  the 
march  of  empire  has  been  along  the  pathway  made  by  the 
public-school  teacher.  He  has  been  the  vanguard  of  progress, 
the  center  of  civilization,  the  harbinger  of  victory,  the  mainstay 
of  liberty.  His  advent  has  always  been  hailed  with  pleasure,  and 
his  presence  has  been  a  source  of  endless  prosperity,  He  has 
toiled  for  the  people  for  centuries  in  a  silence  that  has  been 
golden  for  them,  while  he  has  received,  in  this  world's  goods, 
little  reward.  The  public-school  system  from  its  humble  origin 
has  followed  the  silent  school  teacher,  and  as  silently  done  its 
great  work.  It  has  drawn  from  the  firm  granites  among  which 
it  started,  the  very  elements  of  strength  and  endurance.  Like 
the  sturdy  oak,  it  has  extended  its  branches  over  a  broad  area, 
and  been  a  source  of  admiration  and  joy  for  generations.  Like 
the  spreading  banyan,  its  branches  have  taken  root  in  other 
soils,  without  severing  their  connection  with  the  parent  stock, 
thus  drawing  new  life  and  increased  vigor  from  the  newer  soils, 
while  not  losing  the  benefit  of  the  support  of  the  central  trunk 
that  has  been  its  honor  and  glory.  It  is  a  grand  idea,  this 
confederation  of  states  in  the  educational  union.  In  no  other 
functions  are  the  American  people  so  thoroughly  a  unit,  as  in 
educational  thought.  They  may  be  sectional  and  industrial  as 
regards  tariff  and  free  trade,  partisan  as  regards  state's  rights 
and  centralization  of  power,  sectarian  as  regards  religion;  but 


160  RESUME. 

i 

as  regards  education,  they  are  only  patriotic.  In  short,  the 
educational  system  stands  out,  what  it  is,  the  greatest  American 
institution,  full  of  eloquent  possibilities;  so  grand  and  so  inti- 
mately a  part  of  our  very  national  and  social  greatness,  that  it 
seems  almost  a  reproach  that  in  the  Cabinet  of  the  President 
of  the  United  States  there  sits  no  one  whose  special  duty  it  is  to 
represent  the  educational  interests  of  this  people.  There  should 
be  a  Secretary  of  Education  to  sit  the  peer  of  the  Secretary 
of  War,  the  Secretary  of  Agriculture,  the  Secretary  of  State;  for 
education  is  the  vital  fluid  that  courses  through  and  sustains  the 
life  of  the  State. 

In  the  subjects  to  be  taught,  and  the  methods  of  teaching 
them,  the  teacher  has  his  grandest  field  of  usefulness.  Without 
his  skilled  hand,  little  can  be  done,  particularly  in  the  lattei 
direction;  and  the  subjects  of  mathematics  are,  probably,  receiv- 
ing less,  and  are  in  need  of  more,  attention  to-day  than  any  other 
subjects  in  our  public  school  curriculum.  The  gradual  but  per- 
manent changes  that  have  been  wrought  in  mercantile  methods 
call  for  corresponding  changes  in  instruction  in  arithmetic. 
Many  of  the  subjects  which  are  treated  of  in  nearly  all  text- 
books upon  arithmetic,  might,  with  great  advantage,  be  given 
much  less  time  and  space,  or  omitted  altogether.  This,  to  the  so- 
called  conservative,  will  sound  radical  and  almost  revolutionary. 
But  a  moment  only  of  serious  and  candid  thought  will  be 
required  to  convince  such  that  it  is  in  strict  accordance  with  the 
intent  and  purpose  of  our  public  schools,  ' '  The  greatest  good 
to  the  greatest  number." 

Elementary  geometry  and  concurrent  industrial  drawing  could 
be  taught  with  much  practical  and  refining  influence  much 
earlier  than  they  are  at  present.  These  changes  would  be  in 
line  with  the  best  thought  of  the  world,  and  would  do  much  to 
satisfy  the  increasing  demand  for  manual  training  in  our  public 
schools.  Dr.  Harris  says,  "Industrial  drawing  should  have  its 
place  in  the  common  school  side  by  side  with  penmanship." 
He  further  says,  "Culture  and  taste,  such  as  drawing  gives,  fits 
all  laborers  for  more  lucrative  situations  and  helps  our  produc- 
tions to  hold  the  markets  of  the  world."  Professor  Warren, 
after  pointing  out  in  detail  and  at  great  length  the  uses  and 
purposes  of  geometry,  concludes  by  saying,  "From  all  these 
considerations,  we  may  conclude,  without  rashness,  that  to  not 


RESUME.  161 

less  than  half  of  the  37,000,000  of  industrial  age,  more  or  less 
knowledge  of  geometry,  as  early  and  as  simply  begun  as 
arithmetic  commonly  is,  would  be  highly  beneficial."  This 
number  embraces  more  that  half  the  entire  population  of  the 
United  States;  hence,  the  conclusion  of  Prof.  Warren,  if  it  be 
true,  is  a  strong  one;  and  as  it  was  made  in  the  Forum,  and 
stands  uncontradicted,  it  may  safely  be  quoted  as  not  far  from 
the  truth. 

In  too  many  schools,  however  the  pupil  may  be  taught  at 
first,  as  he  advances,  his  arithmetic  exercises  deteriorate  to  mere 
mechanical  work  on  slate,  paper,  or  blackboard;  never  a  clear 
cut  explanation  from  the  pupil,  or  a  "  why ' '  from  the  teacher. 
The  pupil  is  never  asked  or  encouraged  to  solve  his  problem  in 
another  way,  is  never  encouraged  to  find  a  shorter  method  and, 
therefore,  a  better  one;  hence,  one  of  the  most  effectual  ways 
of  arousing  enthusiasm,  of  begetting  confidence,  of  inducing 
expression  of  thought,  is  not  called  into  requisition.  The 
motto  in  arithmetic  should  be,  "the  shortest  solutions  and  the 
best  explanations."  With  this  end  in  view,  a  variety  of  solu- 
tions should  be  encouraged,'  that  the  pupil  may  be  lead  to  com- 
pare them  and  select  the  best.  In  this  way  a  love  for  the  study 
and  an  enthusiasm  will  be  developed  which  will  certainly  lead 
up  to  a  clear  conception  of  the  subject.  The  very  act  of  search- 
ing for  short  solutions  constitutes  one  of  the  greatest  stimuli  to 
mental  activity  that  can  be  imagined.  It  gives  play  to  genius; 
it  undoes  the  levels  of  the  graded  schools;  it  harms  none;  it 
benefits  all,  some  inestimably.  Explanations  should  always  be 
required  for  the  sake  of  clearness  of  comprehension  of  the 
problem  itself,  for  the  sake  of  the  development  of  the  powers 
of  close  and  exact  expression,  and  for  the  sake  of  building  up 
confidence.  Pupils  need  to  stand  upon  their  feet  and  to  talk  in 
every  study,  so  that  their  language  may  be  criticized  and 
clarified  and  purified.  The  short  solutions  will  result  in  so 
much  time  saved  that  the  number  of  explanations  may  be 
increased  and  their  character  greatly  improved  without  employ- 
ing any  additional  time. 

There  is  also  room  for  great  improvement  in  setting  forth  the 
objects  of  the  study.  Boys,  particularly,  and  all  to  a  greater  or 
less  degree,  are  ever  wanting  to  know  of  what  use  a  particular 
study  will  be;  and,  certainly,  they  ought  to  be  told,  for  it  is 


162  RESUME. 

always  possible  to  do  so.  These  ' '  whys ' '  are  the  germs  of 
knowledge  that  should  be  watered  with  the  dews  of  reason  and 
patience,  that  from  them  may  grow  the  oaks  of  wisdom  and 
mental  power.  Boys  love  the  practical;  they  early  feel  that  a 
great  responsibility  rests  upon  their  shoulders;  and  they  should 
be  encouraged  so  to  feel,  and  assisted  to  prepare  for  their  great 
life  work.  They  should  be  convinced,  as  they  all  can  be,  that 
they  will  not  only  be  wiser  and  better,  but  that  their  chances  for 
making  money  will  be  improved  by  a  thorough  education. 

The  teacher  is  never  without  opportunities  for  improvement. 
There  is,  among  the  many  others,  a  great  work  before  him  the 
faithful  performance  of  which  will  redound  more  to  the  monetary 
and  industrial  advancement  of  the  country  than  it  is  within  the 
capabilities  of  man  to  estimate.  This  work  can  be  done  thor- 
oughly and  well  by  the  teacher,  and  by  him  only.  It  is  a  work 
concerning  which  there  is  a  consensus  of  educated  opinion;  and 
yet  it  seems  no  nearer  accomplishment  to-day  than  twenty -five 
years  ago.  Not  so  near,  in  fact;  for  it  was  then  that  Congress 
authorized  the  use  of  the  Metric  System  of  Weights  and  Measures 
in  the  United  States.  It  was  then  ,that  under  the  stimulus  of 
this  Act,  its  friends  were  led  to  hope,  and  believe  even,  that  its 
introduction  and  use  would  be  easily  and  speedily  accomplished. 
The  real  significance  of  the  law  was  over-estimated.  It  was,  in 
fact,  no  law  at  all.  It  contained  no  compulsory  provisions.  It 
contained  nothing  that  was  tangible.  It  was  like  a  plank  in  a 
party  platform,  pleasing  to  the  ear,  but  possessing  no  legal,  or 
even  moral,  force  within  its  provisions.  The  friends  of  the 
Metric  System  were  lulled  to  silence  and  inactivity  by  it,  and  the 
cause  of  education  in  this  country  was  retarded  in  that  direction 
many  years,  how  many  no  one  can  tell.  The  Metric  System  is 
of  such  incalculable  value  that  no  effort  should  be  spared  that 
will  hasten  its  final  and  universal  use.  It  should,  therefore,  be 
taught  in  every  school  in  the  land;  it  should  be  compared  with 
the  cumbersome  and  illogical  old  system  in  the  presence  of 
every  man,  woman,  and  child  in  the  nation;  its  praises  should 
be  sung  in  the  ears  of  our  people  whenever  and  wherever  oppor- 
tunity offers;  its  advantages  should  be  mathematically  demon- 
strated and  estimated  in  the  years,  months,  and  days  of  a  person's 
life-time,  and  in  the  dollars  and  cents  that  can  provide  so  many 
comforts  and  luxuries.  The  Metric  System  can  be  thoroughly 


RESUME.  163 

taught  in  one  week.  Of  course  it  cannot  be  done  in  so  short 
time,  if  it  is  required  to  change  from  the  units  of  one  system  to 
those  of  the  other;  but  why  should  that  be  taught  at  all?  The 
very  act  of  thus  teaching  it  is  what  has  retarded  its  introduction. 
That  very  mistaken  idea  of  teaching  it  has  made  it  appear  to  be 
beset  with  difficulties,  when  in  fact  it  is  analogous  to  our  money 
system,  the  simplest  in  the  world.  Who  would  think  of  return- 
ing to  our  old  system  of  shillings  and  pence  ?  How  ridiculous  to 
suggest  such  a  thought  even !  Yet,  no  more  ridiculous  than  to 
resist,  or  rather  not  to  encourage,  the  adoption  of  its  comple- 
ment and  coadjutor,  the  Metric  System  of  Weights  and  Measures. 
It  is  the  system  used  by  almost  the  entire  civilized  world.  It  is 
the  system  the  necessity  for  which  John  Quincy  Adams  saw  as 
early  as  1821,  when  he  said;  "Uniformity  of  weights  and  meas- 
ures, permanent,  universal  uniformity  adapted  to  the  nature  of 
things,  to  the  physical  organization  and  the  moral  improvement 
of  man,  would  be  a  blessing  of  such  transcendent  magnitude, 
that  if  there  existed  upon  earth  a  combination  of  power  and  will 
adequate  to  accomplish  the  result  by  the  energy  of  a  single  act, 
the  being  who  should  exercise  it  would  be  among  the  greatest 
of  benefactors  to  the  human  race. ' ' 

The  subjects  of  Square  and  Cube  Root,  Arithmetical  and 
Geometrical  Progressions,  and  Mensuration  will  receive  attention 
in  ' '  Pedagogics  Applied  to  Algebra ' '  and  in  ' '  Pedagogics 
Applied  to  Geometry"  now  in  the  course  of  preparation. 


THE    END. 


I 


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